Sorry, couldn't resist. The only thing I can imagine here is that the paint is offgassing some chemical or other, or the source of the smell is something totally unrelated, like a juniper bush outside your front door.
The question becomes one of economy. How much effort, cussing, and money will it cost to strip the door and refinish it, possibly having it done in a professional's shop, or just replacing the door. This is a question only you can answer. Without knowing what you primed and painted it with, or even the possibility that there is something about the door itself that is either reacting to the finish or something else entirely, without being able to see it or get more of this information from you, this is about all the help I can offer.
Please feel free to post back more specifically, and someone might be able to help you more.