I’m not sure why this question comes up as frequently as it does, but it’s something that seems to be on the minds of many zombie fans.
From a theoretical standpoint, I’d have to say that, yes, they probably do. This is especially true if the zombies are the kind that ravenously feed on human flesh. If the flesh goes in, something would have to come out or the zombie would eventually rupture.
At the risk of being somewhat graphic, the excrement might not be what we normally think of, however. Depending on the level of functionality of the digestive tract, the zombie’s “poop” might just be bits of undigested flesh that simply passed through the monster.
Additionally, a zombie that has had significant trauma to its torso might not have a complete digestive tract remaining. In these cases, the material passing through would likely just drop out from the perforations caused by the wounds. Once again, the level of “digestion” here could vary significantly.
There is a school of thought, however, that says that a zombie might not have the complex muscle control needed to swallow. It could be that they bite and chew, but nothing ever makes it into their stomachs. In this scenario the zombie would not need to poop — nothing in, nothing out.