I've been told that on really humid days, it's actually better to wear the shirt even if it gets drenched. Because the sweat is soaked up by the shirt, the skin is kept drier and cooler. I don't know if this even has any merit, I usually go shirtless anyways. As for sunburn, yeah, you can't really get burnt by running especially if you go in the evening. I've spent entire summers shirtless and wasn't even tan. If you do indeed beleive you can get tan during a run, well then that's another reason to be shirtless since you don't want a farmer's tan.