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"No. Zombie's bodies don't function like living humans. They don't live off eating the flesh of the living, they just have a desire to bite things that are alive. Whatever enters their stomach doesn't get digested, it just sits there."
The Zombie Survival Guide: Complete Protection from the Living Dead (Amazon) asserts that zombies do not retain the digestive functions of living humans:
(p. 11-12) "Recent evidence has once and for all discounted the theory that human flesh is the fuel for the undead. A zombie's digestive tract is completely dormant. The complex system that processes food, extracts nutrition, and excretes waste does not factor into a zombie's physiology. Autopsies conducted on neutralized undead have shown that their 'food' lies in its original, undigested state at all sections of the tract. This partially chewed, slowly rotting matter will continue to accumulate, as the zombie devours more victims, until it is forced through the anus, or literally bursts through the stomach or intestinal lining. While this more dramatic example of non-digestion is rare, hundreds of eyewitness reports have confirmed undead to have distended bellies. One captured and dissected specimen was found to contain 211 pounds of flesh within its system! Even rarer accounts have confirmed that zombies continue to feed long after their digestive tracts have exploded from within."
1) Zombies need nutrition and are not kept alive purely by evil magic
2) Zombies get nutrition more or less the same way the living do, by eating & absorbing nutrients in the stomach.
3) Cranial lesion information from the living can be used to diagnose damaged areas in the undead.
While I'm sure there's a lot of debate in the literary field, this medical student is fairly sure that so long as Zombies ingest matter and gain nutrition from it, they will have to excrete it through their bowels. The vomiting L4D zombies can't have absorbed much nutrition, as there's not a whole lot of absorption in the stomach-- most absorption takes place after you're through the duodenum. This would make L4D zombies INVALID ZOMBIES.
Further, judging by zombie behavior (aggression, ataxia, aphasia, problems with gait and higher-order processing), we might expect to see lesions in the prefrontal cortex (aggression/personality), Broca's and Wernicke's areas (speech production and comprehension), and either the primary/supplementary motor corticies or (more likely) the cerebellum (ataxia, shuffling gait). That zombies don't seem to sleep might indicate a mid-pontine lesion, but that would take away a lot of the higher-order predation functions that most zombies exhibit, so I'd chalk the insomnia up to some thalamic issue-- maybe underactive VLPO.
So we can assume a sick brain. The thing is: the brain isn't totally in control of the digestive system. Distal esophagus and down are controlled largely by the enteric nervous system, which is an incredibly complex processing network (and only modulated by the CNS). Thus, even with a sick brain the zombies would most likely have normal excretory habits, albeit without the conscious sphincter control (CNS) exhibited by most of us living folk.
Your answer, then, is yes. Zombies poop. The probably also pee. It's likely that they're doing these things almost constantly. This isn't to say that the zombies can't have digestive issues that would prevent them from pooping, and I'd get more into zombie gastrointestinal pathology, but that's next year's course and I really don't want to speculate on zombie constipation. Anyway, I hope this helped and please wish me luck on tomorrow's neuro exam.