When I was in Air Force Junior ROTC in high school, we were told you could not have tatoos that were visible while you were in uniform (since they’re not allowed as part of the uniform, while a wristwatch or a wedding ring would be allowed). I’m assuming this would be the same in the military. Most tatoos I’ve seen most guys carrying around tended to be on the upper arm or back, so pretty much any kind of short-sleeved shirt would cover that up handily.
My mom related to me a story of a girl in the Air Force who went out drinking with a bunch of her friends, got blized, and woke up the next morning to find she had a tatoo of a red rose running down her entire arm. She apparantly didn’t get in trouble, but from then on she could only wear long-sleeved uniforms so as to keep the tatoo concealed while uniform.
As for getting court-marshalled for a sunburn, perhaps if you intentionally laid out in your skivvies in a parking lot in Houston during July in order to get a bad enough burn to get out of some unpleasant duty (presumably more unpleasant than a large number of first and second degree burns on your body) then I’m sure they could think of something along the lines of rendering yourself unfit for duty.