I would guess it really depends on the thickness of the paper. The paper pieces I've seen used for hexies is about cardstock weight - I can't imagine that would turn out well in a washer; I'd expect it to get wadded up.
I've heard of people piecing on phone book paper, though, and that's probably thin enough that it'd disintegrate to the point where you wouldn't really notice it; maybe the occasional ball or ridge.
Either way I don't think I'd recommend it!