Does person need to move his lips and his tongue when making Dhikr, Du'a and making Salah (in a time when voice cannot be loud) or it is possible to make it by heart, in his mind only?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The recitation of Fatiha is obligatory in the prayer. During this recitation the person must move his tongue and lips, i.e. the sounds of letters must be uttered from their place of articulation and in their appropriate manner of articulation.
As for what the person has in his mind, it is not called recitation or reading. This rule applies also to the obligatory Dhikr in Salat like Tashahhud and prayer on the Prophet (Sallallahu Alaihi wa Sallam) in the last Raka'h. These two must also be uttered accordingly.
Allah knows best.