Is tape flammable?

I would assume tape is flammable unless it says it isn't.

The shade should NOT get close enough to the bulb to get hot. I recall when most dime store lampshades were made of paper-type material. The tape should be okay, as long as it is not near the bulb.

tape will melt.

Adding tape or paper to the outside of a lampshade is not an issue. Decorate away.

Probably, though probably not in the kind of use you describe, but it might help if you said what kind of tape you are thinking of using. There are at least a dozen things that can all be described as "tape".