One of my players wants to permanently disguise self cause he is a tiefling warlock that has the Mask of Many Faces invocation. If he uses it, will it move along with him as he talks or will it stand still?
Disguise self changes the appearance of the caster
The Mask of Many Faces only gives access to the Disguise Self spell, so the only relevant rules are what the spell itself says:
You make yourself -- including your clothing, armor, weapons, and other belongings on your person -- look different until the spell ends or until you use your action to dismiss it.
(Player's Handbook, page 233. The rules follow up with various other stipulations, none of them relating to speech or movement)
The character's external appearance is truly changed, so it remains in effect with speech and movement. It's not a static illusionary image that just happens to lay over the character.
Since the warlock with the invocation can cast the spell at will, yes, they can remain in their disguised form at all times as long as they spend the action every hour to cast it and can perform the verbal and somatic components.