I have a painted drywall ceiling and want to stick some low-density stuff onto it (things like rope, plastic, maybe some strips of wood). The stuff I stick on would not have a footprint that goes beyond the tape surface, but it might extend a few centimeters down into the room. I want some tape that would stay stuck for a long time with minimal shifting. I'm thinking of an indoor heated room in the US Midwest. Not sure what kind of paint/drywall.
Umm... this is odd. Usually the question is "what kind of tape comes off of drywall"! 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.