Can Quest be run as a MUD or MUSH? And if so, what would I need to do differently?

I know MUDs and IF may seem similar on the surface, but making any sort of game that handles multi-player and syncs up players in real time is completely different from a single player, turn based IF game.

I'd suggest downloading a pre-made MUD codebase to start with, and look into what it takes to host one.