In this instance, the pronoun "who" is the object of the verb "know".
So you want to use objective case whom.
english.stackexchange.com/users/4205/Michael Brown
The quantity of verbs has no effect on the choice between who and whom.
The only thing you need to figure out is whether or not you need a subject for a verb.
If who/whom is the subject of a verb, use who.
Otherwise, use whom.
The non-technical instruction on choosing the right word: Who can only be used as a subject, so if you don’t need a subject, don’t use who; use whom.
For this example, the correct choice is whom.
“The man who/whom I know to be unhappy…” I suppose this is followed by a verb.
The simple subject of the sentence (of the verb that follows) is man, not who/whom.
Who/whom is not the subject of a verb, so you use whom.
Another way to write the example, which might make it easier to parse, is as follows: “The man, whom I know is unhappy,…” As we see, whom is not serving as the subject of any verb.
Source: Precise Edit
english.stackexchange.com/users/11387/Lauren
As the question is tagged with acceptability , I will report the following paragraph, reported by NOAD in the usage of who section: The normal practice in modern English is to use who instead of whom ( Who do you think we should support? )
and, where applicable, to put the preposition at the end of the sentence ( Who do you wish to speak to? ).
Such uses are today broadly accepted in standard English, but in formal writing it is best to maintain the distinction.
If you want to avoid writing who when you should use whom (or vice versa), you can use that .
the man that I know to be unhappy
That is a relative pronoun used to introduce a defining or restrictive clause, especially one essential to identification; it is used instead of when , which , who , whom .
the book that I have bought yesterday the person that I will meet tomorrow the year that Anna was born
english.stackexchange.com/users/252/kiamlaluno
As I write, all other answers agree that whom is the correct choice in this construction (because "whom I know to be unhappy" is an auxiliary phrase, wherein whom is not the subject of a verb).
I don't dispute the strict grammatical position, but I would say that, as suggested by this NGram , whom appears to be increasingly falling into disuse.
Correspondingly, here are over 1000 written instances of " who I know to", most if not all of which are "incorrect" according to strict grammar.
In my opinion, whom is already becoming somewhat 'dated', and it's only a matter of time before it disappears completely.
english.stackexchange.com/users/2637/FumbleFingers
"I know the man to be unhappy" seems be more informative, but it doesn't answer the question until you change it again to "Him I know to be unhappy".
Or how about
"I know that man [him] to be unhappy.
This would indicate that you want the objective case - whom.
english.stackexchange.com/users/4201/jlembke
The main verb in your question is "know", so it is "The man whom I know to be unhappy", just as it is "The man whom I know".
It gets more complex if you replace 'to be' with 'is', as there are several possible meanings.
"
The man who , I know, is unhappy" is equivalent to "The man who is unhappy (I know it)", so whom would be wrong.
"
The man, whom I know, is unhappy" =
"The man is unhappy: I know him (not he).
Without any commas, or (just as wrong)
with a single comma after 'know', ambiguity makes it impossible to say what the pronoun should be (unless the rest of the sentence makes it clear).
Moral:
punctuation is important, and don't lazily cut "to be" down to "is" unless you are clear about how you are changing the meaning.
NB
Precise Edit's answer (quoted by Lauren), leaves out all the commas in this phrase , so isn't helpful.
english.stackexchange.com/users/8019/Tim Lymington
Very simple.
MAN is the subject of the main clause.
The VERB of the main clause is IS.
(This man is unhappy).
"who I know" is a subordinate adjective clause modifying MAN.
The person speaking knows the man.
The subject of the subordinate clause is
I. The verb is KNOW.WHOM is what we call the subordinate clause marker, but it is also the presumed object of the verb KNOW, so it takes the objective
(Whom) case, not the subjective.
When pronouns follow a TO BE verb, and refer back to the subject of the sentence, they take the SUBJECTIVE case.
(I am HE.
She is WHO?)Most of the time, one's ear tells one what word to use.
english.stackexchange.com/users/121389/Lucinda
To answer the direct question ... Verb clauses can be nested and are parsed from the inner most to the outermost.
so "I know '
x to be" is a verb clause which after being parsed would resolve effectively to "is" and therefore your second statement is true - it the correct stereotype verb clause would be "the man who is unhappy".
The response to the other comment - I don't know if this a culture thing
but I do not find it to be correct usage to replace who with that.
That can only be used when talking about NOT-people.
english.stackexchange.com/users/None/
Which of these phrases is more correct?   The man  who  I know to be unhappy  The man  whom  I know to be unhappy   Is one of the verbs in the phrase more important, thus determining the noun case, or is something else happening with the particular combination of verbs? ( The man whom I know  is valid but  the man to be unhappy  isn't; the tense must be specified as in  the man who is unhappy .)
"Who" versus "whom" with multiple verbs
english.stackexchange.com
<grammar>
english.stackexchange.com/questions/10203
english.stackexchange.com/users/4199/Aardsquid
It is generally agreed that, in this instance, the pronoun "who" is the object of the verb "know" and therefore, the objective case "whom" should be used. The number of verbs has no effect on the choice between who and whom. However, it is worth highlighting that "whom" is become dated and it is possibly only a matter of time before it disappears completely. The correct answer is "whom" as the direct object of the verb "to know". The choice between "who" and "whom" has nothing to do with the number of verbs but which noun is the subject and which is the object. "Whom" is the correct grammatical form although its use is waning and may be replaced by "who" in all cases.
It is generally agreed that, in this instance, the pronoun "who" is the object of the verb "know" and therefore, the objective case "whom" should be used. The number of verbs has no effect on the choice between who and whom. However, it is worth highlighting that "whom" is become dated and it is possibly only a matter of time before it disappears completely. In this instance, the correct form would be 'whom', as it is the object of the verb 'know'. However, 'whom' is falling out of usage and 'who' is increasingly common for both subject and object.
The main verb in your question is "know", so it is "The man whom I know to be unhappy", just as it is "The man whom I know". The quantity of verbs has no effect. The correct choice in this case is "whom".
Use whom as object of verb "know" Number of verbs has not effect. Work out subject and object. The whom form becoming data. 
In this instance, the correct form would be 'whom', as it is the object of the verb 'know'. However, 'whom' is falling out of usage and 'who' is increasingly common for both subject and object.
The correct choice in this case is "whom".