Is it necessary to use tape on the exterior of a car when applying wax?

Short answer, NO! With some waxes, there will be a little white residue left in cracks and around trim. This is easily removed with the corner of a rag If you are a real stickler, you can use a cotton swab and/or a toothpick. Many more modern waxes leave no visible residue. One of the big considerations is solvents reacting with rubber seals. Just be a little careful waxing around rubber parts. You'll do fine.If you get any wax on them, just wipe it off quickly.

I do it.

My '02 PT Bruiser has a lot of black rubber door & window seals, plus the black painted bumpers.

If they are not taped, the wax (Carnuba paste, hand applied) gets onto those rubber parts and looks like craap, so it's actually easier to spend the time taping, rather than trying to get the dried wax out later.

Anything not painted or plated (chrome) should be taped, IMHO.

Body shops have a 'wax remover' but it's pricey, and smells nasty. I never found it to be as effective as masking tape.

Make sure to remove tape as soon as the wax is applied, otherwise it tends to leave residue. This is especially true of old tape, or tape that gets wet.

If it is really wax and not some other type of treatment it should go on painted and shiny metal surfaces. It will make a blotchy haze on plastic and painted matte surfaces. Don't put wax where you will not be able to buff it. It does not look good until it is buffed.