If they live long enough, they might. The life time of a zombie is measured in hours, basically. The reason for this is that they are constantly decaying; being 'un'dead, they are risen from the grave, and are already decaying. So the question is: will their GI tracts last long enough to pass the flesh/blood/brains they've consumed, or will their intestines burst first? (" 'E's a poet, and 'e don't know it...")
There are several 'grades' of Zombies, I've found. Shufflers, screamers, runners, sprinters... my studies are continuing. I recently (last night in fact) viewed Day of the Dead (2008), in which it was revealed that SOME zombies actually retain some of their original personalities! (Imagine that one poor kid, who was a vegetarian before he became a zombie! What would he eat?)
The different 'grades' of zombies probably determines their viability in their current climate, i.e. the more active they are the more 'healthy' they are. Shufflers/moaners are almost gone for; sprinters/screamers are likely to be active for several days.
So it is likely that the healthier zombies DO defecate. But they are so covered by other... er... "body fluids..." that it is unlikely to make a difference to anyone they are attacking.
ETA
IN before the "OOHHHH SHIIIT" jokes...