Can We Do Dhikr without Moving The Tongue and Lips?
Question

When we want to say some Dhikrs, is it obligatory to move the mouth? For example, when we want to enter the bathroom and recite Dhikr, should we move our mouths or is it sufficient to say it in our minds? And when we want to go to sleep or say the Adhkar for the morning?
Praise be to Allah.Forms of Performing Dhikr
Remembering Allah (Dhikr) is one of the noblest deeds that a Muslim can do. Dhikr is not only verbal; rather we may remember Allah in our hearts, on our lips and in our physical actions. 
Shaykh `Abd Ar-Rahman ibn Si`di (may Allah have mercy on him) said: 
“The phrase Dhikr Allah (remembrance of Allah) includes everything by means of which a person may draw close to Allah, such as belief (`Aqidah), thoughts, actions of the heart, physical actions, praising Allah, learning and teaching beneficial knowledge, and so on. All of that is remembrance of Allah.” (Ar-Riyad An-Nadirah, p. 245) 
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said: 
“Remembrance of Allah (Dhikr) may be in the heart, on the tongue or in one’s physical actions. The basic principle is remembrance in the heart as the Prophet (peace and blessings of Allah be upon him) said: “In the body there is a piece of flesh which, if it is sound, the entire body will be sound, but if it is corrupt the entire body will be corrupt. It is the heart.” (Narrated by Al-Bukhari and Muslim) What counts is remembrance in the heart, because Allah says (interpretation of the meaning): 
“And obey not him whose heart We have made heedless of Our remembrance, and who follows his own lusts.” [Al-Kahf 18:28]
Remembering Allah with one's tongue or physical actions without remembering Him in one's heart is a serious shortcoming; it is like a body without a soul. 
Remembering Allah in one's heart means thinking about the signs of Allah, loving Him, venerating Him, turning in repentance to Him, fearing Him, putting one's trust in Him, and other actions of the heart. 
Remembering Allah on one's tongue means speaking all words that will bring one closer to Allah, above all saying La ilaha ill-Allah. 
Remembering Allah with one's physical actions means every deed which brings one closer to Allah, such as standing, bowing and prostrating in prayer, engaging in jihad, paying Zakah. All of these are remembrance of Allah, because when you do them you are obeying Allah. Hence you are remembering Allah when you do these actions. Hence Allah says (interpretation of the meaning): 
“and perform As-Salah (Iqamat As-Salah). Verily, As-Salah (the prayer) prevents from Al-Fahsha’ (i.e. great sins of every kind, unlawful sexual intercourse) and Al-Munkar (i.e. disbelief, polytheism, and every kind of evil wicked deed) and the remembering(praising) of (you by) Allah (in front of the angels) is greater indeed [than your remembering (praising) of Allah in prayers].” [Al-`Ankabut 29:45]
One of the scholars said: this means that what prayer includes remembrance of Allah is more important. This is one of the two scholarly views concerning this verse.” (Tafsir Surat Al-Baqarah, 2/167, 168) 
Can we do Dhikr without moving lips?
With regard to the Adhkar which are spoken on the tongue, such as reciting Quran, saying Tasbih (Subhan Allah – glory be to Allah), Tahmid (Al-hamdu Lillah – praise be to Allah) and Tahlil (La ilaha ill-Allah – there is none worthy of worship but Allah), and the Dhikrs to be recited in the morning and evening , when going to sleep, when entering the washroom, and so on, it is essential to move the tongue, and a person is not regarded as having said them if he does not move his tongue. 
Ibn Rushd (may Allah have mercy on him) narrated in Al-Bayan wat Tahsil (1/490) that Imam Malik (may Allah have mercy on him) was asked about a person who recites when praying, but no one can hear him, not even himself, and he does not move his tongue. He said: “This is not recitation, rather recitation is that in which the tongue moves.” 
Al-Kasani (may Allah have mercy on him) said in Bada’i` Al-Sana’i` (4/118) 
“Recitation can only be done by moving the tongue to say the sounds. Do you not see that if a worshipper who is able to recite does not move his tongue, his prayer is not acceptable? Similarly he swears that he does not recite a Surah from the Quran but he looks at it and understands it but he does not move his tongue, then he is not breaking his oath.”  This is because he has not recited it, rather he has only looked at it.  
This is also indicated by the fact that the scholars said it is not allowed for a person who is Junub to recite Quran with his tongue, but they said it is permissible for him to look at the Mus-haf and recite Quran in his heart, without moving his tongue.” 
This indicates that there is a difference between the two things, and that not moving the tongue is not counted as reading or reciting.  (See Al-Majmu`, 1/187-189)
Shaykh Ibn `Uthaymin (may Allah have mercy on him) was asked: Is it obligatory to move the tongue when reciting the Quran in prayer or is it sufficient to say it in one's heart?
 He replied: 
“Recitation must be done with the tongue. If a person recites it in his heart when he is praying, that is not sufficient. The same applies to all other Adhkar; it is not sufficient to recite them in one's heart, rather it is essential to move one's tongue and lips, because they are words to be spoken, and that can only be achieved by moving the tongue and the lips.” (Majmu` Fatawa Ibn `Uthaymin, 13/156)
For more details, please refer to the following answers: 258651 , 139841 , 194998 , and 10491 .
And Allah knows best.