Short answer
Remembering Allah in your heart is rewarded, but the spoken adhkar, such as Subhanallah after salah or the morning adhkar, only count as said when you move your tongue and form the words. In the Hanafi school the safer view is that you should be able to hear yourself. Silent dhikr of the heart is good alongside it.
For any dhikr, dua or recitation whose reward is tied to saying it, the words must be pronounced with the tongue; thinking them is not saying them. The minimum per al-Karkhi is correct pronunciation of the letters, and per al-Hindwani, audible to oneself, which many later Hanafis preferred.
The four schools agree on this.
Explanation
There are two kinds of dhikr. Dhikr of the heart is being mindful of Allah, reflecting on His blessings and His names. Allah says, "remember your Lord within yourself in humility and in fear, without being apparent in speech" (7:205), and in a hadith qudsi, "if he remembers Me within himself, I remember him within Myself". This is rewarded and is the root of all dhikr.
Dhikr of the tongue is saying the words. The specific adhkar of the Sunnah, such as 33 Subhanallah after salah or Sayyidul Istighfar in the morning, are words to be said. The scholars of the four schools agree that running them through your mind is not saying them, just as reading the Quran with your eyes is not reciting it: see reading the Quran in your head.
How quietly is enough? You do not need to be loud. Moving your lips and tongue to form the words, at a whisper that you yourself can hear, is enough and is the safer Hanafi view. You can do this on a bus, at your desk or in a queue without anyone noticing.
The best is to join both: words on the tongue with attention in the heart. If you are in the bathroom or another place where speaking dhikr is disliked, remember Allah in your heart only. See the best dhikr and the dhikr guide.
The evidence
Quran, Surah Al-A'raaf 7:205
وَٱذْكُر رَّبَّكَ فِى نَفْسِكَ تَضَرُّعًۭا وَخِيفَةًۭ وَدُونَ ٱلْجَهْرِ مِنَ ٱلْقَوْلِ بِٱلْغُدُوِّ وَٱلْءَاصَالِ وَلَا تَكُن مِّنَ ٱلْغَٰفِلِينَ ﴿٢٠٥﴾
And remember your Lord within yourself in humility and in fear without being apparent in speech - in the mornings and the evenings. And do not be among the heedless.
Inner remembrance is praised.
Sahih al-Bukhari 7405
Sahih... فإن ذكرني في نفسه ذكرته في نفسي ...
"If he remembers Me within himself, I remember him within Myself."
Remembrance in the heart is rewarded.
Jami' at-Tirmidhi 3375
Grading to be checked... لا يزال لسانك رطبا من ذكر الله ...
"Keep your tongue moist with the remembrance of Allah."
Dhikr of the tongue is a distinct act to keep up.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Salah, discussion of the minimum level of quiet recitation. The views of al-Karkhi and al-Hindwani on what counts as speech.
- al-Adhkar (Nawawi), Introduction, on dhikr of the heart and the tongue. The prescribed adhkar count only when said so that one hears oneself.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.