The moon will speed up. This is due to the tide acting as a brake between the earth and the moon. The earth rotates at 1 rev per day, the moon at 1 rev per month. The brake effect will cause the moon to speed up and the earth to slow down. Eventually the moon will be fast enough to leave earth orbit. Thankfully, I won't be around when that happens. The rather surprising fact is that tidal energy does not come from the moon at all but instead comes from the earths spin.
Mike.
edit, actually the moon won't leave earth orbit, it will stop accelerating when the earth and moon rotate in unison once every 47 days!! **broken link removed**