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When a door is opened and closed repeatedly, it's no surprise that they can start to tarnish. Scratches and other superficial damage to the hinge can create friction and cause the door to stick. If you are worried about the lubricant possibly staining the door, you can take the additional step of removing the door from the frame. You can find out how to remove a door from the hinges here. Just like the lubricant can stain the door, it can stain your floor. Cover the area directly under the hinges you are working on to prevent any dripping. After you've done this, open and close the door a few times to work the jelly into the hinge joints. WD-40 would also have the same helpful effect on your door, but it is messier and dries out faster.
Decide if the door hinges are causing the trouble. Lay newspaper on the floor. Apply petroleum jelly to the hinges.