Is it a Ford Taurus? Have you checked the coolant level? Did you check the coolant hoses right at the fire wall?
Probably not the door itself but the motor that works it. They are prone to fail. You can get access to it by removing the cover below the glove box and removing the radio. The blend door motor is usually white in color. It will have an electrical connection plugged in it.Turn the blower on low and slowly turn the blend knob. you can usually feel and hear the air change over if the door is moving any at all. If not you can remove it from the box, plug it in, and see if it works that way. Look at where the motor plugs in the blend door itself. Check for cracks and make sure the door moves freely. If it is broken you will have to remove the dash and replace the blend door.
Hope this helps you out. If so please accept the answer. If you have any more questions just ask.
Ford Technician
9 Years of Professional Ford Experience. Current Shop Owner.
You have to have a ford radio removal tool. You can pick these up a radio shack, wal mart, parts stores. They stick in the holes on the side of the assembly. push till they click and pull them gently away form (pull one left and one right) each other while pulling out on the radio. Hard to explain. I drew some arrows in the pic. Sometimes it can be tricky to get the clips to disengage. The motor is not real hard to remove. Be careful not to break the linkage below it behind the metal shield you ll have to remove. Plug the same connector in and see if it moves while turning the knob from hot to cold. If not replace it. Dealer will usually have one in stock. I haven't had much trouble from the doors on the Taurus. If it is busted and not busted too bad you might be able to rig the motor and door up to stay open. But it mounts flush and might be tricky to do.