I have hunted some and I always thought it wise to stand back at a prudent distance and make sure the animal is, indeed, expired before getting too close. Like the original MacGyver used to say, "You never know."
My concern was never getting too close, it was always not being able to see it if it did manage to get up and take off. Around here, the woods are mostly pretty thick and you can't see far from the ground. Most all of the long shots I've made were from tree stands or hilltops/sides so once you start walking toward that deer, you have almost no chance of seeing it again until you're right on top of it. Deer, whitetails at least, don't tend to play 'possum. If they can still move at all, they will struggle to get up immediately or with very little delay. Still, after dropping one, I always kept the safety off and my sights on it for a good 10 minutes or so before even thinking of headin' that way.