Is it obligatory to vocalize when reciting the Quran or performing dhikr, or is merely moving the tongue sufficient, or is it permissible to recite and remember God with the heart without moving the tongue? And which of these is best?

The tongue must be moved when reciting (Qira'ah), performing remembrances (Adhkar), and supplications (Du'a) during prayer (Salah); otherwise, it is not considered a recitation. However, outside of prayer, it is permissible to perform remembrance with the heart without moving the tongue, though it is preferable to combine both.