I'd recommend "ran out the door" if you are referring to a person or animal moving from inside a building to ouside via a door, for example: "He ran out the door". It would be better to use "ran out of the door" to describe the movement of something from the inside to the outside of the door itself. " However, in informal AmE, you will hear both "he ran out the door" and "he ran out of the door" used to mean the same thing.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.