When one's throat and sinuses feel like crap, and the back of their throat gets just that way so as to make swallowing hurt but not in the conventional sense, but in a way that I can only describe as a feeling of grossness combined with a dull pain, their voice tends to sound a certain way.
Well, assnose is when your voice sounds that way when you *aren't* sick. Basically like you're trying to put the least possible amount of effort on your nose when you're talking, so it ALL comes out the mouth instead.
Etymology: When the deepest parts of your nose feel like ASS! (except they don't, you just sound that way)