You have to wait for the fire to burn a certain amount of time. You will know it’s readywhen it is giving off sparkles. If you waited long enough and don’t see sparkles, you can check by dropping a villager on the waterfall on the right. If he collects water in a bowl and dumps it on the fire pit, you can solve this puzzle. If he just gets a drink, you’ll have to wait longer.
When the fire is out, you will see a white diamond gem on the fire pit. Immediately afterward, it will be too hot for your villagers to pick up. You can either have them pour bowls of water on it, or just wait for time to pass. When the gem is cool, drop a villager on it and he will take it to the door.