Will a person be rewarded for remembering Allah or reciting the Qur'an to himself without moving his lips, outside of prayer?

Reading without moving the tongue and lips is not considered recitation, because it is merely thinking and does not entail the legal rulings of recitation, such as breaking an oath or preventing a ritually impure person (junub) [from touching the Qur'an].

As for the reader receiving reward for reciting in their heart, that is among the matters of the unseen. However, this type of reading is considered an act of worship if it is accompanied by contemplation of meanings, contentment, and submission. Allah the Almighty has commanded contemplation of the Noble Qur'an, for the Qur'an was revealed for its verses to be pondered and its meanings to be reflected upon, not merely for recitation.