The Moon already produces tides in both the Oceans and the Earth itself. If something with a significant amount of energy where to hit the Moon, there are a few possibilities:
1. Moon shatters, but the resultant pieces do not have enough energy to overcome the gravitational binding energy of the Moon. The pieces separate and the then fall back together to reform a moon-massed object. The Moon's orbit will likely be changed.
2. Moon shatters with enough energy to it from reforming but not enough for them to escape the Earth's gravity. The pieces end up forming a ring around the Earth.
3. Moon shatters with enough excess energy to knock the pieces completely free of the Earth entirely.
The tidal effects caused by case 1 depends on the new orbit. If the collision alters the orbit so that on average it is closer to the Earth, we would get larger lunar tides, the closer to the Earth it gets, the stronger the tides. For them to be Armageddon scale would require it to get really close.
Case two would have the tides caused by the Moon weakening as the ring formed. We'd still have solar tides. This could take a fare bit of time, so I don't see this as being a sudden change. The Earth is already used to being flexed on a 12 hr or so cycle, so I don't see it reacting violently to having this flexing weakening over time.
With case three, the Lunar tides would weaken as the debris field gets further from the Earth. Again this will not be fast, and not cause much stress on the Earth.
So really, only case 1, where the collision robs the Moon of enough orbital energy to give it a close perigee would produce greater tidal forces. I think it would be difficult for such an impact to have enough energy to lower the perigee significantly without imparting enough energy to shatter the Moon into a ring.