Burning sensation on skin of upper arm/shoulder for 2 weeks, but nothing there, skin looks completely normal. No rashes or other symptoms. What is it?

Neuropathy. Sounds like a nerve problem, so we'd call it neuropathy. It can be caused by anything damaging a nerve. Shingles or herpes can do this, but if it has been going on for 2 weeks I'd expect there to be spots or bumps or something showing on the skin by now. It can also be caused by your immune system attacking the nerve or by the nerve being pinched in your spine or elsewhere. See your doc for help.