Q:

What factors bear on the authenticity of Matthew 23:14?

Modern translations of the New Testament omit some verses found in the King James Bible (missing verses). The reason for the differences is that modern translations use the Nestle-Aland edition of the Greek New Testament (the critical text) where verses have been removed from the received text on which the King James was based.
This issue is seen as a decision on the quality of the documents used to make the translation. The Nestle-Aland edition uses manuscripts which were not available to the King James translators. It is based on the view that the Alexandrian-text type manuscripts are better manuscripts. Verses not found in these manuscripts which are present in the Byzantine type manuscripts are seen as scribal additions to the original text and in an attempt to present what scholars believe is the most accurate original text, the suspect verses are left out. 
While manuscripts which differ in the number of verses could be explained by either adding or removing something when making a copy, variances between manuscripts are considered as scribal additions and the suspect verses are excluded from the critical text. When a change is made, the verse numbers are left unchanged and the “missing” verse may be noted in a footnote.
Matthew 23:14 is one of the removed verses:

Woe unto you, scribes and Pharisees, hypocrites! for ye devour widows' houses, and for a pretense make long prayer: therefore ye shall receive the greater damnation. (KJV)

As the full chapter in Matthew consists of Jesus rebuking both the scribes and the Pharisees it presents the opportunity to use different methods to evaluate if this verse was present in the original. It also provides the opportunity to assess the affect of removing a verse from a passage and why that verse (and not another) was omitted.
Is there evidence and/or arguments (beside of the quality of manuscript) to support a conclusion that Matthew 23:14 was present in the original text and intentionally removed?

A:

Textual Witness Analysis
Here is what the Apparatus to NA28 (which omits the verse) indicates:1

ουαι δε (− 700. 892c) υμιν γραμματεις και Φαρισαιοι υποκριται οτι
  κατεσθιετε τας (− Δ) οικιας των χηρων και (− 1424) προφασει μακρα
  προσευχομενοι δια τουτο ληψεσθε περισσοτερον κριμα ƒ13 it vgcl syc
  bomss (p. vs 12 K W Γ Δ 0102. 0107. 565. 579. 700. 892c. 1241. 1424. l
  844  f syp.h bopt)   
¦ txt א B D L Z Θ ƒ1 33. 892* a aur e ff1 g1
  vgst.ww sys sa mae bopt

NOTE: The "p. vs 12" indicates many of the witnesses testifying to the reading have the verse after v.12. 
Both the omission and the inclusion have nearly equally ancient Greek readings:

Omission, א = Codex Sinaiticus (4th c.)
Inclusion, W = Codex Washingtonianus (4th/5th c.)

Both the omission and the inclusion have broad translation readings early (only indicating the two variants emerged early, such that early translations could pick up from each variant):

Latin (Used in Mediterranean Countries of Rome, primarily Italy)

Omission, a = Vercelli, Bibl. Capitolare (single manuscript, 4th c.)
Inclusion, it = All or a majority of Old Latin witnesses (3rd-4th c.)

Syriac (Used in the Fertile Crescent and Eastern Arabia)

Omission, sys = Syrus Sinaiticus (4th/5th c.)
Inclusion, syc = Syrus Curetonianus (5th c.)

Coptic (Used in Egypt; Bohairic was a northern dialect)

Omission, bopt = Five or more Bohairic witnesses support the particular reading (3rd c.)
Inclusion, bomss = Two to four Bohairic witnesses support the particular reading; For after v.12, bopt = Five or more Bohairic witnesses support the particular reading (3rd c.)

Only inclusion is supported by majority texts in Greek and most early translations, though there is some split in the Coptic; and it is unclear what is the actual "majority" reading there, except omission is testified in the mae (middle Egyptian) and a (southern Egyptian) witness in conjunction with the bopt (partial northern Egyptian witness):

Greek,  = readings supported by the majority of all manuscripts
Latin, it = All or a majority of Old Latin witnesses (3rd-4th c.)
Syriac, syp.h = Peshitta and Harklensis, the former being "the most widely accepted of the Syriac versions."

Conclusion on Textual Witness
Disclaimer: I'm essentially a Majority Text person. Nevertheless, when I find...
Early Witness + Widespread Witness + Majority Witness

... and assuming the reading is not superseded by a Super Majority Witness or clear contextual logic against it,2 then I do not even bat an eyelash at considering the Majority reading to not be original. 
It then comes down one's other factors. For me, the Majority Text logic makes more sense, so I follow that, and it is all I need.3
Internal Evidence
Dick Harfield's answer has a paragraph summarizing internal arguments for inclusion, three points noted there I believe are significant:4

"scribes and Pharisees" phrasing
"hypocrites" usage
The seventh woe point, but in reverse. There are seven Woe's noted with respect to the phrase "scribes and Pharisees" (if v.14 is included), but it actually makes eight Woe's total, because v.18 says "woe" to "the blind guides," which is in context obviously still a reference to the scribes and Pharisees. The exclusion of one Woe to make it into seven would seem to be more likely a possible purposeful omission by a later copying scribe to force a preference for seven (merely conjecture here and I'm not one to make charges of "heresy" against the Alexandrian texts or their copyists; the texts play an important role as well in textual criticism, and the copyists just as much prone to accident as any in copying).

However, more internal evidence of inclusion can be brought forth. Recall that the majority of witnesses have v.14 after v.12, with v.13 following both, so as the The New Testament in the Original Greek: Byzantine Textform 2005 shows:

12 Ὅστις δὲ ὑψώσει ἑαυτόν, ταπεινωθήσεται· καὶ ὅστις ταπεινώσει ἑαυτόν, ὑψωθήσεται.
13 [v.14 in TR] Οὐαὶ δέ ὑμῖν, γραμματεῖς καὶ Φαρισαῖοι, ὑποκριταί, ὅτι κατεσθίετε
  τὰς οἰκίας τῶν χηρῶν, καὶ προφάσει μακρὰ προσευχόμενοι· διὰ τοῦτο
  λήψεσθε περισσότερον κρίμα.
14 [v.13 in TR] Οὐαὶ ὑμῖν, γραμματεῖς καὶ Φαρισαῖοι,
  ὑποκριταί, ὅτι κλείετε τὴν βασιλείαν τῶν οὐρανῶν ἔμπροσθεν τῶν
  ἀνθρώπων· ὑμεῖς γὰρ οὐκ εἰσέρχεσθε, οὐδὲ τοὺς εἰσερχομένους ἀφίετε
  εἰσελθεῖν.

Which would be in English (using the NKJV with ordering to the Majority Text as above, rather than the minority testified that the mislabed Received Text uses):5

12 And whoever exalts himself will be humbled, and he who humbles
  himself will be exalted.
14 [v.13 MT order] Woe to you, scribes and Pharisees, hypocrites! For you devour
  widows’ houses, and for a pretense make long prayers. Therefore you
  will receive greater condemnation.
13 [v.14 MT order] “But woe to you, scribes and Pharisees, hypocrites! For you shut up
  the kingdom of heaven against men; for you neither go in yourselves,
  nor do you allow those who are entering to go in.

The importance of this is that internally, the majority witness inclusion and ordering makes more sense to the flow of the passage, not only between v.12-14, but through all the Woes to v.30.
Transition into and content of Woes 1 thru 3

V.12 is a statement about humility, and how those who do not have it will be humbled—an allusion to judgment.
V.13 (MT order) is the 1st Woe directed against the scribes and Pharisees (S & P), and it is directly against their lack of humility, placing themselves above widows and doing a show of prayer. The warning is for a greater judgment. 
V.14 (MT order) is the 2nd Woe against S & P, indicating the judgment involves not entering the kingdom of heaven, and that they block other's from entering.
NOTE: The placement of v.13 per MT order makes more sense, as it transitions the logic an thought smoothly from the statement of v.12 to v.14.
V.15 is the 3rd Woe and reads (NKJV; all further quotes from this version):

Woe to you, scribes and Pharisees, hypocrites! For you travel land and sea to win one proselyte, and when he is won, you make him twice as much a son of hell as yourselves.

The charge is that they actively seek and make others into worse persons than themselves even, but also indicates the ultimate destination of "hell."

Woes 4 and 5
The topic of the first three Woes is this intensification of judgment. The next two Woes mention nothing of judgment, but Christ is just "teaching" the S & Ps where how their "making" of others worse than they comes about. 

V.16-22 gives the 4th Woe, but changes in that it refers to the S & P crowd as "blind guides":

Woe to you, blind guides, who say, ‘Whoever swears by the temple, it is nothing; but whoever swears by the gold of the temple, he is obliged to perform it.’ 17 Fools and blind! For which is greater, the gold or the temple that sanctifies the gold? 18 And, ‘Whoever swears by the altar, it is nothing; but whoever swears by the gift that is on it, he is obliged to perform it.’ 19 Fools and blind! For which is greater, the gift or the altar that sanctifies the gift? 20 Therefore he who swears by the altar, swears by it and by all things on it. 21 He who swears by the temple, swears by it and by Him who dwells in it. 22 And he who swears by heaven, swears by the throne of God and by Him who sits on it.

So Christ is demonstrating that they have no understanding of truth about reality. What they teach is a messed up set of priorities, and this infects others. 
V.23-24 the 5th Woe connects the "blind guides" to the S & Ps, in case someone thought he was talking of another group (I say that sarcastically; that it was the same group should be obvious to anyone reading):

23 Woe to you, scribes and Pharisees, hypocrites! For you pay tithe of mint and anise and cummin, and have neglected the weightier matters of the law: justice and mercy and faith. These you ought to have done, without leaving the others undone. 24 Blind guides, who strain out a gnat and swallow a camel!

Both the 4th and 5th Woes covers monetary issues, but leading up to correcting them about what some of the real "weightier matters" are—the spiritual points of justice, mercy, and faith.

Woes 6 thru 8
The final three Woes balance with the first three, which focused on their prideful show and then external actions, whereas these last three are focused on internal motivations and again a prideful show. The connection between the external focus to internal focus was made through the monetary vs. real spirituality teaching of the 4th and 5th Woes.

V.25-26 is the 6th Woe, reading:

25 Woe to you, scribes and Pharisees, hypocrites! For you cleanse the outside of the cup and dish, but inside they are full of extortion and self-indulgence. 26 Blind Pharisee, first cleanse the inside of the cup and dish, that the outside of them may be clean also.

This is purely a rebuke of the "inside" issues of pridefully seeking exaltation. Note how "extortion" parallels "devour widows’ houses" of the 1st Woe (v.13 MT order) and "self-indulgence" parallels "for a pretense make long prayers." So a topical match between 1st Woe and 6th Woe of heart issue manifesting in action. This is internal testimony both to the inclusion and the position of v.13 in the MT order.
V.27-28 the 7th Woe reads:

27 Woe to you, scribes and Pharisees, hypocrites! For you are like whitewashed tombs which indeed appear beautiful outwardly, but inside are full of dead men’s bones and all uncleanness. 28 Even so you also outwardly appear righteous to men, but inside you are full of hypocrisy and lawlessness.

Again, issues of weightier matters such as hypocrisy and lawlessness dwelling in their hearts is the topic. There is a picture parallel contrast between the 2nd Woe (v.14 MT ordering) and the 7th. The kingdom is shut up from being entered by the S & Ps, whereas they have shut themselves into the sealed tombs of their sinful hearts. This is internal testimony both to the inclusion and the position of v.14 in the MT order.
V.29-33 then reads for the 8th and final Woe:

Woe to you, scribes and Pharisees, hypocrites! Because you build the tombs of the prophets and adorn the monuments of the righteous, 30 and say, ‘If we had lived in the days of our fathers, we would not have been partakers with them in the blood of the prophets.’ 31 “Therefore you are witnesses against yourselves that you are sons of those who murdered the prophets. 32 Fill up, then, the measure of your fathers’ guilt. 33 Serpents, brood of vipers! How can you escape the condemnation of hell? 

Like the 3rd Woe, the 8th parallels it with an activity of the S & Ps. In Woe 3, they sought converts for their future continuing of their ways. In Woe 8, they (hypocritically) seek connection to the past prophets of God as their supposed spiritual forefathers. There is also a parallel in that as the S & Ps made converts worse than themselves, they are themselves going to "fill up, then, the measure of your fathers' guilt," i.e., they are going to do worse than their fathers, though they claim they would have not done what their fathers did in persecuting God's messengers. The final parallel of Woe 3 to 8 is the mention again of hell (or better, Gehenna, which is reference to the lake of fire). These final parallels in Woe 8 to Woe 3 are internal testimony corroborating the parallel ideas of v.13-14 (Woes 1 & 2) being intentional to v.25-28 (Woes 6-7), as v.15 (Woe 3) has these continued parallels to v.29-33 (Woe 8).

Conclusion on Internal Organization
The inclusion by the Majority Text of Mat 23:14 is testified to by internal evidence. The proper positioning of it is different from the earlier Received Text analysis, which positioning makes a proper transition in logic flow for the passage as well as proper parallels within the structure of the argument through the eight Woes.
Historical Situation and Further Synoptic Literary Analysis
An examination of the Synoptic Gospels themselves shows that the statement in question (v.14 TR, v.13 MT) occurred in the same historical setting to which all three Gospels (Matthew, Mark, Luke) are testifying. This is in some way more or less recognized by every one of the six harmony of the Gospels I examined.6 Such a recognition is to be expected, as the context of all three testifies clearly to the relation (NOTE: the EOC refers to "end of chapter").

╔══════════════════════════════════════╦════════════════╦═══════════════╦════════════╗
║                Event                 ║    Matthew     ║     Mark      ║    Luke    ║
╠══════════════════════════════════════╬════════════════╬═══════════════╬════════════╣
║ Question about Son of David          ║ 22:41-45 (EOC) ║ 12:35-37      ║ 20:41-44   ║
║ Discourse Against Scribes/Pharisees  ║ 23:1-36        ║ 12:38-40      ║ 20:45-47   ║
║  (a) long robes                      ║ v.5b           ║ v.38          ║ v.46a      ║
║  (b) market greeting                 ║ v.7            ║ v.38          ║ v.46a      ║
║  (c) best seats                      ║ v.6            ║ v.39          ║ v.46b      ║
║  (d) widows' houses & long prayer(?) ║ v.13 (MT)      ║ v.40          ║ v.47 (EOC) ║
║ Widow's Mite                         ║ n/a            ║ v.41-43 (EOC) ║ 21:1-4     ║
║ Lament for Jerusalem                 ║ 23:37-39 (EOC) ║ n/a           ║ n/a        ║
║ Prophecy of Temple Destruction       ║ 24:1-2         ║ 13:1-2        ║ 21:5-6     ║
║ Olivet Discourse                     ║ 24:3ff         ║ 13:3ff        ║ 21:7ff     ║
╚══════════════════════════════════════╩════════════════╩═══════════════╩════════════╝

Observations

The preceding and following context shows that the historical events recorded by each are in a roughly parallel location to each other if Matthew had included the v.13/14 reference at (d). That is, the Discourse Against Scribes/Pharisees occurs in the same location in relation to the Question about Son of David and Prophecy of Temple Destruction. This testifies to the same historic event being the focus of all the passages.
If Matthew included v.13/14 reference (d), it also falls parallel correctly to sometime after the other three parallel concepts of (a)–(c). Only the ordering of (a)–(c) shows any distinction, but considering Matthew gives much more detail in the section on Discourse Against Scribes/Pharisees, Mark and Luke are likely giving a summary. The fact that Matthew gives more detail of the denunciations than Mark are Luke at this event testifies to a higher likelihood that he would not have omitted such a denunciation from his Gospel. The location of the (d) following the (a)–(c) for all three additionally testifies that if Matthew included it, it should be located sometime after those other statements, which is true for both TR (v.14) and MT (v.13) reading locations.
The Majority Text reading locating the statement at v.13 (1st Woe), besides being more internally consistent (see "Internal Evidence" above), offers the first half of a better explanation on why Mark and Luke only recorded that statement. Often when people are listening to speakers, the speaker may say something and then dwell on that thought, tuning out what immediately follows. So if this was the first statement of Woe as the MT order testifies, then it would more likely be remembered. The second half of explanation on why Mark and Luke only recorded that statement is found in context of their Gospels. This statement was important because their focus was immediately to turn to the Widow's Mite event, an event Matthew chooses not to record. The statement of devouring widow's houses had literary value to set up the event of this contrast of a widow to Pharisaic giving. Their extremely shortened summary of denunciations and literary purpose to tie to the Widow's Mite gives a reason for their omission of other denunciations, and further testifies to Matthew being the more detailed of the three regarding the denunciations, and thus less likely to have omitted it.
Matthew's focus in his gospel was upon the condemnation of the scribes and Pharisees, and so in place of the Widow's Mite, his literary set up is gearing toward the prophetic judgment portion to come, and so gives the Lament for Jerusalem. As the "Internal Evidence" showed, the MT location in v.13 as 1st Woe has literary value for bridging between v.12 and v.14, but in this parallel examination, we find it also had literary value in bridging to the Lament for Jerusalem. Consider the parallel:

v.13 (MT, 1st Woe): you devour widows’ houses
v.38 (Lament): See! Your house is left to you desolate

The concept forms an inclusio of ideas around in which Jesus begins with what the Pharisees have done to widows' houses and ends with what will happen to the house (the Temple) the Pharisee's so loved because of their works. This literary connection to the Lament for Jerusalem that only Matthew chose to include testifies to v.13 (the Majority Text location) as being original.

Conclusion on Historical Events and Further Synoptic Literary Points
The historical event order, literary purposes of each Synoptic Gospel writer, and literary structure of Matthew's argument testify to its being original to Matthew and to why the other two Gospel's only mentioned it of all the points made in Matthew's Woes.
Conclusion
The evidence from (1) external, (2) internal, (3) historical, and (4) synoptic literary purposes all point strongly to INCLUSION of v.14 (TR) immediately after v.12 (as v.13), followed by the "traditional" v.13. per Majority Text reading.7

NOTES
1 Eberhard Nestle and Erwin Nestle, Nestle-Aland: NTG Apparatus Criticus, edited by Barbara Aland, Kurt Aland, Johannes Karavidopoulos, Carlo M. Martini, and Bruce M. Metzger, 28th revidierte Auflage [revised edition] (Stuttgart: Deutsche Bibelgesellschaft, 2012).
It should be noted that one Wikipedia article also indicates a few other Uncial manuscripts not noted in the NA28 Apparatus apparently have the reading:

Matthew 23:14 ... 0104, 0107 [only this one noted in NA28 app.], 0133, 0138

2 Super Majority Witness is my term to indicate that spelling errors, certain grammatical issues, etc., even if they ended up in the majority of manuscripts, are testified to in the broader Greek corpus (outside the NT) as being incorrect. By "clear contextual logic," I mean an internal evidence that would make the writer look like a fool if it had been written that way, because he would be contradicting himself elsewhere in the context. Personally, I am not familiar enough with some variants to know if any majority text testify to such an internal conflict, but if they did, that would give me pause to possibly look at a non-majority witness.
3 This post cannot get into nearly all the points, but by "Majority Text logic," I mean:

It is widely acknowledged that most of the variants arose in 2nd-3rd centuries (e.g., "the period of the greatest variation, the second and third centuries," Jeff Kloha, "A New Edition of the Greek New Testament," Concordia Theology Journal, 2012), the period of the least amount of textual witnesses, and the witnesses we do have primarily from a single locality—Egypt, because it was dry and conducive for preservation.
We are then missing all the early witnesses that must have existed in Asia Minor (near the Byzantium copy centers) during this early period. "Must," because the Byzantine tradition did not just erupt from nothing. But even more significantly, copies in that area would have more likely been accurate to most of the NT. Why? Because 74-89% of the original manuscripts (i.e. destinations to audiences) of the NT were written to the areas near Byzantium: Asia Minor and Eastern Greece. So during those formative years when variants were occurring the most, that area of the world would have had the easiest access to cross-checking to most of the original texts, which most likely did still exist those first 100-200 years in that area.
So I find statements like Bruce Metzger's comment in A Textual Commentary on the Greek New Testament, Second Edition a Companion Volume to the United Bible Societies’ Greek New Testament (4th Rev. Ed.) (London; New York: United Bible Societies, 1994) on v.14, "its absence in the earliest and best authorities of the Alexandrian and the Western types of text," unhelpful because it ignores the copy testimony from the most important region. And I find statement's like the NET Bible's note 17 on the verse, "The most important mss ... do not have 23:14" (emphasis added), inaccurate at best and deceptive at worst. All the manuscripts are important as witnesses, including the later date manuscripts that carry on a memory of an earlier reading from which they were copied.

This is what I mean, in part, by Majority Text logic that I find more compelling than mere "earliest" logic. We are missing too much of the "earliest" testimony to place much weight on that.
4 I would disagree with Dick Harfield on at least two points in that paragraph:

I would argue Jesus did use some form of ὑποκριταί. The Hebrew חנף is the root used that translates sometimes into English as hypocrisy or hypocrite. It means to be godless or profane (BDB) or irreligious (HALOT). But it is used in contexts of hypocrisy: Bildad's charge against Job (Job 8:13; also later Zophar 20:5), a usage Job acknowledges himself in 13:16 (though in defense that he is not one), and in Job 27:8-10, Job is again using the term in context of one who calls on God at times, but obviously is like the wicked. So Hebrew has the concept available in a single term, especially given context.
I hold Matthew to be earlier than Mark. Some call this a Matthean priority, but that assumes the gospels did borrow from one another (or some lost source), and my view of the inspiration of the text (i.e. God-breathed; it is His word as much as it is the human author's) does not require that (though it may be that sources were used).

5 I say Received Text is "mislabled" because the Majority Text is really the Received Text of the churches. That is, the reading that testifies to being in the majority was what was "received" by the most churches and used the most in history. So the work to refine the Majority Text reading is simply, in my view, a continuation of refining the reading for the Textus Receptus that began back in the 16th century.
6 The six I examined were:

Kurt Aland, Synopsis of the Four Gospels (Bellingham, WA: Logos Bible Software, 2009), n.p., §194.
Ernest DeWitt Burton, A Harmony of the Synoptic Gospels for Historical and Critical Study (New York; Chicago; Boston: Charles Scribner’s Sons, 1917), n.p., §§156–158.
Steven L. Cox and Kendell H. Easley, Holman Christian Standard Bible: Harmony of the Gospels (Nashville, TN: Holman Bible Publishers, 2007), 168, §137.
Benjamin Davies, ed., Baker's Harmony of the Gospels (Grand Rapids, MI: Baker Book House, 1994), 120-122, §122-123.
Jeffrey Glen Jackson, Synopsis of Matthew, Mark and Luke (Bellingham, WA: Logos Bible Software, 2009), n.p., under "Woes to Pharisees and Scribes."
A. T. Robertson, A Harmony of the Gospels (Bellingham, WA: Logos Bible Software, 2009), n.p., §137.

7 Some may wonder what my conjecture is on why/how the reversing of order occurred in the minority of witnesses that have the older Received Text reading/ordering. Considering we do not possess the vorlage or the copy in which it first occurred, any speculation is conjecture. One thought is that copying was stopped at v.12 and picked up again at a later date, but when done, what has become v.13 was copied first, but the error was caught, so the copyist decided as long as all the Woes were put down, what did the order matter? So he copied what has become v.14 next. This copy then became a vorlage for the family of copies that testify to the reversal. The switch was more than likely an earlier error (i.e. perhaps non-professional scribes) on a copy specifically sent out to another community since it made its way into a number of the earlier translations. It seems most likely, however, that the switch was accidental, as I cannot think of a reason to purposefully do so.
Regarding possible reasons for removal:

I already noted one possible intentional reason it was removed, that is to make the Woes balance to seven (a mystical number for many people with respect to Scripture). 
Another intentional reason would be knowledge by scribes that two versions did exist (v.13/14 TR order, versus v.13/14 MT order), and so either removed it (a) fully, thinking it an addition, or (b) partially, the initial vorlage that had it omitted may have left a gap for later inclusion if some further investigation was going to be done to see if the reading was deemed valid, but then copyists of that gapped vorlage did not know the reason for the gap and so left no gap.
A possible accidental reason is similar to the switch above. Since the majority text shows that the omitted verse was the first of the eight Woes, then if someone left off copying after v.12 and picked up copying later, since there are eight Οὐαὶ (Woe!) words, they may have accidentally missed copying the first one. How? If the first one was at the bottom of a leaf (or last on a section of a scroll), then perhaps when copying was resumed, the next section/leaf was started with, forgetting (or not knowing if a different scribe) that the former copy session left off before the start of the Οὐαὶ phrases, rather than at the end of the section (i.e. left the Οὐαὶ sections to be copied as a group, even though the first one may have been on the preceding section of scroll).
Another accidental scenario, similar to the above. If the first Οὐαὶ was at the end of a section, and that section was damaged (i.e. missing), then the last seven Οὐαὶ may have been copied simply because they survived in that vorlage.

One can conjecture for hours on possible scenarios for both switch and exclusion, but just like those who advocate for omission, all conjectures on why it might have been added (i.e. gospel harmonization) are just that, conjectures. All that we know is that three main variants exist, with some minor variations in those as well. How they came to be is lost to history.

A:

The Idea in Brief
Very able Bible scholars in years past have addressed this question. Both the United Bible Society 4th Edition Greek New Testament (UBS4) and the Nestle-Aland 28th Edition (NA28) indicate that the verse in question would not appear to have appeared in the original versions of the text. There appear several reasons for this conclusion.
Discussion
One of the most well-known and popular Greek New Testament scholars with the United Bible Society was the late Bruce Metzger. Metzger observed that the best witnesses to the Gospel of Matthew had omitted the verse. He wrote the following in this regard, and he ascribes the highest confidence rating {A} given to the opinion of the United Bible Society.

23:14 omit verse {A}
Verse 14 is not included in the earliest and best manuscripts of the Alexandrian and the Western types of text. Copyists have clearly added it from the parallel text in Mark 12:40 or Luke 20:47; this is confirmed by the fact that some copyists added it before v. 13, and others added it after v. 13.

But exactly how did Metzger and others arrive at their highest confidence rating? According to the most recent and updated critical apparatus available in the Novum Testamentum Graece (28th Ed.)(NA28), the following notes appear:
 
The brackets [ ] around the [14] indicate that the verse is omitted by the editors of the NA28, and in this regard, the witnesses highlighted in pink reflect the omission of verse 14. On the other hand, the witnesses highlighted in yellow are in favor of verse 14 appearing in the gospel account, but please note the particular witnesses in the parenthesis indicate that the text in question comes not after, but before verse 13.  
In addition, there are at least five church fathers whose witness also bear upon this issue. The following diagram provides the visual depiction of witnesses within the first five centuries. Bohairic and Syriac witnesses do not appear in this image, because they appear in all variations (that is, they both omit and include the verse in question), and therefore provide no relative diagnostic value. The following diagram therefore reflects Latin and Greek witnesses.
Please click to enlarge.

Origen lived during the late Second and early Third Century in Alexandria, Egypt, and therefore had direct access to one of the greatest libraries of the ancient world before its subsequent destruction after his death before the end of the Third Century.
For example, Origen appears to have been aware of variant readings of the Gospel of Matthew. When Origen wrote his Commentary to the Gospel of John, he made explicit and declarative reference to the “Second Woe” in the Gospel of Matthew. Please click on the image in order to view the source (Book XX §352, lines 10 ff).   
The translation follows.

 
Furthermore, it will also be possible to prove something from opposites which would seem to be paradoxical. Now the paradox is this, that someone is more a son of God than is another son of God, and that someone is twice as much a son of God as another. 
We will demonstrate how this is proven from its opposite as follows. The second “woe to the scribes and Pharisees” in the Gospel according to Matthew is as follows. “Woe to you, scribes and Pharisees, hypocrites, because you traverse sea and land to make one proselyte, and, when he is made, you make him twice as much a son of hell as yourselves.” [bold emphasis added]

If Origen had access to the library in Alexandria during his 70 years, then he may have had direct access to variant readings of the Gospel of Matthew within that library or variant readings known by other Christian scholars frequenting Alexandria. (For example, Clement of Alexandria was the protégé of Tatian shortly before Origen was born in the city of Alexandria.) In the diagram above depicting the Greek and Latin witnesses, the red arrow would indicate that Origen was writing his declarative comments concerning the “Second Woe” in contradistinction to other witnesses such as Tatian, which provide variant readings. (Bruce Metzger also notes in his UBS4 Critical Apparatus that the commentary of Origen appears in both the Greek and Latin versions.) In other words, Origen was providing textual criticism to his Greek and Latin readers concerning the correct reading of the passage in Matthew, which, in the opinion of Origen, had omitted the verse in question in the Gospel of Matthew.
Finally, as already noted at the beginning of this discussion, the highest confidence {A} assigned by Metzger and his colleagues stemmed from the Alexandrian sources and Western witnesses.  On Page 15 of his Critical Apparatus, Metzger provides the following observation concerning weight assigned by the United Bible Society to the witnesses:

This citation indicates that the United Bible Society considered both Clement of Alexandria and Origen of Alexandria (in part) on equal par with witnesses such as א and B as primary Alexandrian sources. Please note that the editors of the United Bible Society do not include W as a primary witness. In other words, א and B in addition to Origen (in part) are given more weight as witnesses than to W, which was the first uncial attesting to the inclusion of the verse in question. In addition, the same citation indicates that the United Bible Society assigns more weight to D as a Western Witness than to W, which they regard only as primary for certain portions in the Gospel of Mark.
Conclusion
In summary, the United Bible Society with the late Bruce Metzger, in addition to the editors of the Nestle-Aland NA28, had made the correct observation that the text in question (verse 14) does not appear to have been part of the original Gospel of Matthew. In this regard, two immediate reasons were because (a) Origen appears to have made the declaration and explicit distinction concerning the “Second Woe.” In other words, Origen was providing textual criticism to his readers concerning the passage in contradistinction to other variants extant at the time and known by Origen; and (b) the inconsistency of the other witnesses to the contrary, which favor the inclusion of this verse. That is, these witnesses are divided concerning whether the verse in question comes before or after verse 13. This inconsistency raises more questions than answers assuming that the verse were original to the autograph by Matthew. 
For these two salient reasons, the text in question (verse 14) does not appear to have been part of the original transmission of the Gospel of Matthew.

References: 
Nestle, E., & Nestle, E. (2012). Nestle-Aland: NTG Apparatus Criticus. (B. Aland, K. Aland, J. Karavidopoulos, C. M. Martini, & B. M. Metzger, Eds.) (28. revidierte Auflage). Stuttgart: Deutsche Bibelgesellschaft, 76.   
Omanson, Roger L. & Metzger, Bruce M. (2006). A Textual Guide to the Greek New Testament: an Adaptation of Bruce M. Metzger’s Textual Commentary for the Needs of Translators. Stuttgart: Deutsche Bibelgesellschaft, 41.   
Origen (1993). Commentary on the Gospel according to John Books 13–32. (T. P. Halton, Ed., R. E. Heine, Trans.) (Vol. 89). Washington, DC: The Catholic University of America Press, 268.

A:

Matthew 23:14 is absent from some earlier manuscripts, which is a clue that it was not in the autograph, but not actual proof of this.  David E. Garland (The Intention of Matthew 23, pages 15-16, footnote) says the evidence against its inclusion is strong, including text type and broad geographic base, while the evidence for its inclusion is weakened by textual variants.
As I indicated in my answer to this question the use of the Greek ὑποκριταί makes it unlikely that these words were literally spoken by Jesus. The author was writing in Greek and may simply not have realised that there was no directly equivalent Aramaic word - no one can suggest that anyone could recall Jesus' exact words decades after he spoke them (of course, it is possible that Jesus said something more or less similar, although not a literal equivalent). By itself, this does not mean that 23:14 was not original to Matthew. As we will see in the next paragraph, Matthew's use of this word could actually be evidence that 23:14 was original to the text.
In favour of the inclusion of Matthew 23:14 is that Mark 12:40 does refer to the scribes devouring widow's houses, as does Luke 20:47. We know that Matthew's Gospel was substantially based on Mark (1), so this correspondence makes it somewhat more likely that 23:14 is authentic - some 90 per cent of Mark's material found its way into Matthew, because the author of Matthew generally did not omit material he found in Mark unless he had reason to do so.  Also somewhat in favour of its inclusion is that Mark 12:40 and Luke 20:47 only criticise scribes, but Matthew 23:14 follows vv13,15,23,25,27,29 in referring to "scribes and Pharisees," thus a peculiarly Matthean theme. Of course, if verse 14 was an insertion, the editor could have copied a repeating theme of "scribes and Pharisees." In the same way, both 23:14 and 23:15 refer to hypocrites, using the Greek word ὑποκριταί, so it seems that whoever wrote Matthew 23:14 also wrote 23:15. Garland says (ibid, page 18) that verse 15 should be considered a Matthean composition because of the high number of Matthean words, when compared with material sourced from Mark and Q. Perhaps most conclusively, verse 14 contains one of seven woes, and without this, there would only have been six: As Garland points out in a different context, the author seems at pains to have exactly seven woes.  Although not conclusive, this points to Matthew 23:14 being part of the original published version of Matthew's Gospel.
Footnote

(1) Nearly all New Testament scholars accept Mark to have been a source for Matthew. John Dominic Crossan, in The Birth of Christianity, page 111, calls this a massive consensus. Adam Winn says (The Purpose of Mark's Gospel) that in the first eight years of this century, at least eight significant critical commentaries on Mark’s Gospel have been published; all eight assume Markan priority as a starting point. Of course, a massive consensus does not always prove a point, but in this case, there seems to be clear internal evidence for Markan priority.

