strangely it's the tidal movements that are causing the moons orbit to increase, it's kinda a gravitational friction between the tides and the moon that tries to push the moon further forwards in it's orbit and at the same time making the orbit slightly larger.
The increase won't go on for ever though, there will come a time (eventually, if the earth is still in existence ) when the increase will stop with the orbit being about 1.6times what it is now.
as for the tides, even if the moon ceased to exist right now we would still have tides that would be caused only by the gravitational pull of the sun, but they would be about a third of the size we get at the moment.
However if the moon had never existed at all it would be a totally different story, in the same way that the gravitational friction is pushing the moon forward in its orbit it also slows the earths rotation down a tiny bit, if there had never been a moon we would have days that would only last about 6 hours which would have very extensive effects on life on earth.