You don't need to wait until it burns out for itself. You can distinguish the fire with water, then cool the gem with water, and take the gem.
In my first tribe, I wondered what the sparkles meant for a long time, but I saw the gem only after I accidentally forgot to feed the fire. But for the 2nd, 3rd and 4th tribes I didn't wait that long; I put off the fire right after spotting the sparkles, and it worked fine.