Irene:
Normally, the tumbler inside a dryer won't be "painted", it'll be powder coated. Powder coating is where they electrostatically spray paint the tumbler with a mixture of powdered plastic and tiny colored particles (called "pigments"), and then bake the result in an oven. During the baking process, the plastic melts and bonds to the underlying metal, encasing the coloured pigments in that clear plastic much like raisins are encased in the bread in raisin bread. Powder coatings are usually very impermeable, and so it's not likely the ink will actually penetrate in to "stain" the coating of that tumbler, so you should be able to get the ink off completely with an appropriate solvent.
I don't think the heat from the dryer will make the ink any harder to get off. Inks are made to dry fast, and dryers don't get very hot, so there isn't going to be much change in the situation as a result of leaving the dryer off or using it now, several days after the ink stained the dryer.
You might want to throw some old clothes in that dryer and let them tumble when hot to see if any ink comes off on them. If push comes to shove, it's not hard to replace the entire tumbler in an US made electric dryer, but I don't know how much a new tumbler for a dryer costs. Try dissolving the glue with some solvents or nail polish remover first, tho.
