Heh, urine is sterile, that's a fact. And I've heard the same thing.
Not sure I'd recommend it...but I have heard it.
Seajoy, now that the burn has stopped burning, you can use a little ice to bring down any swelling. Take an aspirin (or tylenol if you're cursed like me and can't take any NSADS) and keep the hand elevated. (that helps with the pain) If there are any blisters, DO NOT pop them. Let them be because the liquid inside helps with the burn. If they pop on their own, put nesporin on them and call your doctor.
When I worked in the ER, we'd use a bit of Dreft laundry soap to soak burns in to help them stop burning. It's a very very gentle soap. Dreft, cool water and then a bit of ice in the water once the burn had stopped. (burns keep burning even when you pull away from the heat.)
I sure hope it feels better soon!
Kae