What would the result be of re-painting a slightly rusting door?

The most expensive but probably the best fix is to replace what you have with a rust-free used door and have them repaint it when they do the other side.

Second best would be to media blast the rust on the door with crushed walnut shells. Treat the rust after media blasting with a rust preventative. Do any patching on the outside that's needed with filler and have them refinish. It should buy you a few rust free years if it's done right.

Repaired rust ALWAYS comes back. Always. The galvanized coating that was placed on that part of the body when the car was built has eroded away, and paint alone will not prevent the rust.

There is no magic cure that will stop that from happening unless you just replace the door. Even then, it could be a pattern failure (i.e. all cars of that make and model rust there) in which case replacing the door just buys you time.

As several others have indicated, you need to address the rust prior to getting it painted, otherwise the rust will be back and will probably be worse. There are several videos on YouTube that show how to attack and stop the rust before you get it painted. It will take some elbow grease, some simple tools and some consumables such as primer, filler, rust converter, etc. It's not a hard job, more time consuming than anything else, but above all, don't paint over it.

Here, check this out, https://youtu.be/n4vusY2-rkQ

I'm going to be rattle canning my truck, I've done this before. I came across a great DIY video by ChrisFix on YouTube. Here is link to the video: https://youtu.be/vUdSUDObwVc , check out 12 minutes into the video, Chris talks about hiding "hard lines" even if just wanting to paint a small section of the door, usually the entire panel will need to be done, atleast until you can find a seem or edge to blend the existing paintjob with the new.

Hope this video helps!