He also looked like he had some sort of dental prosthetic that made him look like Val Kilmer to me--his teeth seemed to protrude more than usual. Either that, or it was how he held his upper lip. He certainly looked different as Ennis, though, without any prosthetic, so it could just be good physical acting.
You're right, Meryl! I've also wondered if the teeth are fake. And the Val Kilmer comparison is perfect; now that you mention it, I was subconsciously thinking the same thing.
The real-life version of Heath's character, Skip Engblom, appears in the special features, and he has teeth like that. (Interesting side note: Heath and the real Skip are interviewed together, and Skip claims to have been the one who insisted on Heath playing him in the movie, though it's not clear why.)
EDIT: Not clear why because he doesn't say, that is, not because it's not obvious why someone would want to be portrayed by Heath.