When I've checked in to hotels that were fully booked on the day of my planned departure, I've been asked to sign forms confirming the check-out date and waiving my right to stay on. If they do that, they're probably (IANAL!) within their rights to kick you out. Other than that, the worst they can do is probably raise your rate to the legal limit.

(That may be a small part of why the posted rates inside the doors of so many U.S. hotels are so far above even the rack rate, though other factors such as the ability to charge that much on the local university's homecoming weekend, Mardi Gras, or whatever, surely factor into that as well.)