Thanks for all the tips!
One of the problems was that I hadn't scuffed the surface before applying the primer, so the adhesion of the tape was evidently greater than that of the primer to plastic.
I've been using some 3-M "painter's tape", and because of the fact I'm doing the hull of a battleship, the width of the tape is about 2 inches, which helps cover larger areas but does seem pretty sticky. I'd call it more of a "blue" color rather than a purple, and if there are different colors for different strengths of adhesive, I need the less sticky stuff.
I was doing some more work last night, and as a "workaround" I simply used my hands on the sticky side to take some of the adhesive off before I applied it to the model and that seems to work. I do seem to need to be a bit more careful with the edge when there's less adhesive, but at least it's not pulling off the paint that's already on there!
Thanks!!!
Tom