It's hard to do some sorts of genres in tabletop RPGs, and one of them is horror. (The other is romance.) The common player is sufficiently detatched from the situation (as one expects in a medium of play derived from tabletop wargames) that the sort of visceral reaction that proper horror produces in the audience--dread, terror, fear, etc.--are absent in tabletop gaming. This is why tabletop RPGs have to resort to employing mechanics to generate the same effects.
Perennial favorite Call of Cthulhu has its SAN mechanic to enforce the Cosmic Horror feeling of Lovecraft's stories, despite the fact that what we know now of the cosmos renders so much of that moot. Yet, to keep that feel, you got chuck the dice and watch out for your mental Hit Points as much as you do your physical ones. Otherwise you get play experiences so contradictory to the source material that you lose the point of the game. (This is why the Cthulhu boardgames are better experiences; they dispense with playable spaces outside of what the sources allow.)
Horror requires scaring the audience to succeed, and tabletop RPGs don't allow for that. That's the horror of horror RPGs: it's like being a eunuch- you still dig it, but you can't do it, so there's no payoff and therefore no reason to bother trying without bringing in widgets to workaround the dysfunction. Tabletop RPGs are about doing, not being, and doing trumps being psychologically; this is why "just do it" is actually sound advice most of the time.
So, with all this crap going on, what known tabletop RPG does its source material well without needing Fuck You mechanics used on players? Well, that's easy, and if you can find a copy of the rulebook I recommend giving it a read:
All Flesh Must Be Eaten
It's not strictly a zombie-themed horror game. It's an adventure game where the zombie rules are applicable for damn near any sort of monster that fills that niche: zombies, vampires, xenomorphs, terminators, slasher movie killers, etc. (And yes, that means you can replay Cabin In The Woods if you like.) The sample settings and supplementary materials make this crystal clear. Yes, there is a PDF available, and you can dig for that on your own if you (sensibly) avoid SocJus-converged outlets like RPGNow and DriveThruRPG.
If you're unwilling or unable to use Call of Cthulhu or All Flesh Must Be Eaten, and you want something more blatantly using that tried-and-true D&Dish framework, well there's one go-to choice that's been in print for a generation now: Beyond The Supernatural. Palladium also has its own zombie RPG (Dead Reign) and Hidden Monster PC game (Nightbane) if you want those flavors of horror on said framework.