There are actually two answers to this question and it really depends on the hotel...

It is not uncommon for guests to stayover without formally letting the hotel know. It really depends on hotel policy and state law as to what they can and eill do. I would not bank on being able to keep the same rate for the second night. It is very unlikely that they will kick you out, especially if you have a credit card on file.

The other side of this is that they may formally ask you to leave, but then again, refer to part 1...