Most hinges on interior and exterior doors have removable pins, allowing you to take the door down without unscrewing the hinges. The pins aren't removable on some other types of door hinges, including the spring-loaded, self-closing door hinges you may have on your storm door. When that's the case, you do have to unscrew the hinges to remove the door.
Removing Hinge Pins
Close the door to prevent it from falling when you remove the pins.
Look underneath the bottom hinge. If you see a small hole, insert a 2-inch common nail through the hole and tap up against it with a hammer to force out the pin. By the time the nail is all the way into the hole, you should be able to pull the pin out with your fingers. If not, extract it with pliers.
Wedge a flat-head screwdriver between head of the pin and the hinge if you don't see a hole under the hinge. Tap the screwdriver with a mallet or hammer to dislodge the head, then increase the angle of the screwdriver and keep tapping to force the head up. Extract the pin with your fingers or pliers.
Repeat the procedure with the top hinge. Open the door carefully and lift the door off the hinges.
Close the door.
Unscrew the screws holding the bottom hinge to the door frame, using a Phillips screwdriver. Spring-loaded hinges are always spread out like a butterfly and mounted flat on the outer frame.
Unscrew the top hinge from the frame, then open the door and remove it. Lay the door on a flat surface and unscrew the hinges from the door. Put the hinges and screws in a safe place.