just pull that stuff off with your bare hands. I did mine this past weekend. came off really easy on my old '91.
Maybe the tape was a little broken down from being so old or maybe that os just how the stuff is.
When i took mine off the tape kinda split, half onto the door and half on the trim. after you get the trim off just use a hair dryer to heat the tape up a little and work it off. I used a old puddy knife that had been dulled out around the edges and the left over tape came off pretty quick. Be carefully if you use a edgde of some kind, you could easily scratch the paint. It will come off using your fingers just might take a little longer.
After you get the tape off just clean it and you should be good to go.
I wasn't to concerned with what the paint was like under the trim when i took it off, so i may have gone a little faster than you want to. I am planning a complete repaint in a couple weeks once i wrap up school for the quarter.
i tried the fishing line thing when i started and i think it hurt my hands more than the tape holding the trim on. Thats when i decided to just grab it and start pulling it off. Make sure you move your hands along with the trim as you pull it. If you get to far away from the door with you hands the pull won't be as controlled and you might buckle your door or something. You will hear it before it goes.
Point is once you have it started it will come off easier than you think. Slower you go the easier it will be to keep everything underneath looking good. Clean it with some kind of solvent that won't hurt your paint once you get all the tape off and you should be all set