I think that your question is very thought-stimulating, sherminator. Let's imagine just making the moon disappear rather than blowing it up.
Well, jake-the-peg is quite right about the common centre of gravity shifting and that would probably kill all life on earth and indeed shatter the earth itself just by the fact that it would move several hundred miles (just a guess) outside its angular momentum to the new centre in just a few hours. We'd all be squished!
Also due to the moon's gravity, the oceans bulge out toward the moon and there is an anti-bulge on the side away from the moon resulting in the tides. All the water in each bulge would suddenly be let go resulting in a tidal wave that would swamp the earth continuously for a very long time, probably years. That would mean extinction for us as well. As if that weren't enough the hot, internal magma of the earth also feels the gravity of the moon so the result would probably be smashed-up continents and lava flows everywhere as the magma did the same thing as the oceans.
Sherminator, please don't blow up the moon or make it disappear!
The ramifications of your interesting thought experiment have really set me thinking, so thanks for that. Whilst you were upset by the flippant answers, I have to say that albaqwerty's comment about the werewolves had me in stitches!
Regards
Peter Jameson