For this to even be a thing that happens there would need to be something to fuel the fire; furthermore any debris that would "fall" from the moon would most likely burn up in our atmosphere, in fact it's even more likely that if the moon were to break apart it would simply create very many small moons.
If all of the Helium3 within the moon were to combust in some freaky fireball resulting, somehow, in the moon exploding: there would be a considerable danger for gigantic chunks of the moon hurtling towards the Earth, the lack of a moon would cause our tides to be decided by the Sun (which would only be 40% as pronounced), nights would be EXTREMELY dark, the lack of a gravitational pull from the moon would speed up the rotation of the Earth considerable (resulting in a day to only last 6-8 hours), and an assortment of other terrible things.