When you have a flat tire or any other damage to your car, simply drive out to the countryside (so that a loading screen appears). After the load, your car will be fixed.
In Free Ride mode, head out of town if you get into trouble. There are three exits. Two are in Chinatown, and one is the way to the airport. This will repair any damage to your car, get police or gangsters off your tail, and if you die it will be your last saved point. This will also clean up all the weapons that are left lying around. Unfortunately it will also remove the cars that you left in town. It will not heal you or fill your car with gas. It will stop your car from losing gas quickly if your car is damaged. Note: This does not work in the small town.