Is this a good idea for sticky lips or should I avoid
I would assume that it might gunk up the holes on the harmonica, but then I am only new to the instrument, so maybe someone with more experience than myself can give you a more definitive answer.
My lips often become very chapped, particularly in the winter. I use lip balm as needed with no problems.
Yeah, I don’t really see any problem with it. If you are using the deep relaxed mouth position, your lips should be deep on the cover plates, so it shouldn’t really have any way to get inside the holes and muck up the reeds.
I’d say, if it feels better to play with it, go right ahead!