It's unknown if they do, but it's likely. Headcrabs gain control of the host's organs one by one, eventually taking full control. Considering zombies are only truly killed by killing the attached headcrab, it is to be presumed that the only thing keeping the host alive is the headcrab.
Addition: This may or may not be the case. When listening to the standard Zombie's sound files backwards results in incredibly disturbing cries for help and of pain in a shrill, but (relatively) clear human voice.