This is a tough one... if the moon was blasted out of Earth orbit by asteroid impact, there would be many things to consider.
First, the effect of the asteroid, which would really need to be a planetoid. Anything big enough to dislodge the moon would surely perturb the Earth's orbit before hitting the moon. It might be dragging smaller asteroids with it (see Armageddon!) which could impact the Earth and wipe out much of life. The moon could be broken apart by such an impact, but for the sake of the question I'll assume it stays mainly intact.
Certainly the collision itself would send massive rocks and tonnes of dust our way - again, planetary fireballs, tsunami, nuclear winter and such would wipe out huge swathes of life. Any remaining life could suffocate or freeze or starve to death.
Then, there's the effect of suddenly not having a moon. The Earth and moon actually are a double-planet, due to the size of the moon compared to Earth. We both orbit a common point, which will change (think of spinning round holding on to a kid, then letting the kid go - you'll suddenly be off-balance). The exact effect of this will depend on the speed the moon is moving away from us at and it's exact path. It's safe to assume the result will be some slight alteration to the Earth's orbit around the sun, whether it is the radius or eccentricity (measure of 'oval-ness' of the orbit) that is slightly different. This would change the amount of sunlight reaching the Earth, or at least affect the seasons. The surviving life could be burnt from too much heat and sunlight or get frozen to death and any life that depends on the now-dead lifeforms (for food, for example) would also die.
Where the moon goes could keep having an effect on the Earth. If it is hanging around in an orbit that intersects Earth's there could be further changes to the Earth's orbit as the moon and Earth pass near each other.
The tidal changes would be nearly impossible to predict. No more tides would happen (unless the moon passed close by) which would affect the flow of the oceans etc.
All in all, it's gloomy and improbable, just like Space:1999.
