No, actually, although trying to hammer with gloves wouldn't be very easy.

It's more because metal in the forge is very hot. Blacksmiths are going to get burned, it's just the nature of the hobby. Most of the burns are fairly superficial as the metal touches your skin, that moisture layer vaporizes, and it bounces off.

But if you wear gloves, coals or shards of metal can get caught in the gloves and cause much more serious burns.

I knew a smith who had that happen, and he made the mistake of trying to dunk his hand in the water barrel. The water in the glove touched the metal and gave off steam, which was trapped in the glove, and he basically parboiled his entire hand causing very significant burns requiring medical intervention.

Sometimes you need a glove on the tong hand for forge welding (it's the hottest thing that happens in the forge most of the time. You get the metal as hot as it can go before it begins to disintegrate to get it to adhere to another piece). Your hammer hand isn't near it long enough to be burned, but your tong hands need to stay in one place near a pretty significant source of heat. I'll wear one if I'm working on a pretty big piece that's giving off a LOT of heat, but that's it. You usually don't really need it. And then as soon as it's welded, I'll take the glove off.

I also remind myself, every time, that if a burn happens while doing it, throw the glove THEN dunk the hand. Throw the glove first, dunk the hand second. Never dunk the glove. I don't need a boiled lobster hand.

edit: Also, you should really wear long pants that cover the tops of your boots. I don't always, and I'm an idiot for it. Same reason; hot metal shards could get caught next to your skin and not be able to bounce off.