Depends on the hotel. Most around the airports I've left cars at (LHR, IAD and RIC) charge either a Park & Fly rate and/or get you to sign a waiver form so you can leave it there. Sometimes I've had to put a little dangly thing off the rear view mirror, sometimes not.

Why not just take a taxi to/from the airport? Unless you live miles from Tallahassee that's got to be cheaper than staying at an airport hotel or leaving the car at the hotel.