What sort of tape adheres best to painted drywall?
Umm... this is odd. Usually the question is &quot;what kind of tape comes off of drywall&quot;! Darn near anything will stick to painted drywall far better than you'd want it to.

Masking tape will likely hold reasonably well, but may not support much weight and will dry out and fall off after a couple of years.

Duck tape (the generic cloth-backed stuff, not specifically the brand name) will hold a reasonable amount of weight and will last for years.

Duct tape (the aluminum/foil backed stuff specifically designed for duct work) probably won't hold as well as it does to to galvanized steel ducts

Gaffer's tape will hold about as well as duck tape and will, generally, come off cleanly, but will leave marks after a couple of years. (Ask me how I know)

Electrical tape probably won't hold all that well or support all that much weight.

Double-sided carpet tape will probably do the job quite nicely especially if the wall is pretty smooth.

Double-sided foam-backed tape may hold pretty well, especially if either the surface of the wall or item to be mounted is rough/textured.