I know that a reversible process does more work than an irreversible process and that extra work is turned into heat. But what does that heat do? Does it increase the temperature?
The heat is dissipated and lost to the universe forever. This is why it is impossible to create perpetual motion machines.
The viscous dissipation of mechanical energy that occurs in irreversible processes involving deformation of a fluid is typically referred to as "viscous heat generation." But this is a bit of a misnomer because it is typically manifested as an increase in the internal energy of the system (so it is not, strictly speaking, heat). Of course, this change in internal energy can be reduced (or added to) by transferring heat from (or to) the system to (or from) its surroundings (across the system boundary).