Depends upon whether you're allergic or not. I've been moving my RBTAs barehanded for years, not to mention cleaning my tank barehanded. I've noticed that the burning sensation I get on the back of my hand has increased, but I'm not sure if that's due to an increasing allergic reaction on my part, or a stronger sting on the part of my RBTAs.
In any case, it feels kinda like a nettle sting - a burning/itching sensation, but it goes away with a good hand washing, and never leaves any marks. So, unless you're one of the few who have an allergic reaction, your BTAs sting shouldn't cause you any problems.
The above being said, there ARE many reasons to wear gloves while banging around in one's aquarium. I've had a microbacter marinum bacterial infection on my knuckles, and it's not much fun.
FWIW,
Kevin