One of the graduate schools I'm applying to asks if I would like to be considered for admission even if they can't fund me. Would it hurt my chances of receiving funding if I checked "yes?"
It shouldn't but it might.
For example, in my department, if you are the best candidate then you will get the funding no matter what. However, if you have exactly the same qualifications with another candidate (almost never happens) and we have funding only for one, we would offer the other candidate funding and to you to come without funding. Of course, once at the department, we would try to get funding for you for the next semester/year.
All these, depend on the department policy though.
That very much depends on the local policy. In any case, if I had funding for, say, 5 students and could take on 10, I'd fund the 5 best (or the 5 best which asked for funding), and got 5 out of the rest. But that's me...