How close are you usually to the deer? Like 100 yards? Or is that crazy that you would be that close.
I don't personally deer hunt, but know quite a few people who do. I'm just getting into hunting myself, gearing up for coyote and squirrel. Deer hunting in Illinois with a firearm is quite a pain, as it is only shotgun, muzzleloader, or handgun. The few firearm deer seasons there are are short, and even getting a deer tag can be luck of the draw. No rifles allowed, the supposed reason being that the land is to flat and densely populated so shots could carry and be dangerous (which might make sense if not for there being absolutely no restriction for coyote, but I digress). All that being said, there are also archery seasons, and you definitely get close to them then.
So, with that preface out of the way, it all depends on where you are hunting. Here in Illinois, where we are more limited in what we can use, 200 yards is getting to the edge of the capability of rifle barreled shotguns with sabot slugs and muzzleloaders can humanely take deer sized game reliably. With a smoothbore shotgun and slugs, 100 yards is more realistic. Out to 100 yards, open sights are certainly feasible if not always ideal. Good shooters could very well reach out beyond that, but I wouldn't rely on it. They sell shoot barrels with rifle style sights built in for shotguns specifically for that kind of hunting.
In states with less restrictions and more open space, hunters with .308 can take deer as far as 700 yards and even a bit beyond that, at those distances obviously a scope would be necessary. More powerful cartridges could extend that range even further out, I just went with .308 because it's one of the most common.