I know that if you add mass to the moon, it would get closer to the Earth. We all know that the moon causes the tides because it's gravity pulls the water. So, my question is: If the moon gained more mass and got closer to the Earth, could it have enough pull on the water that it actually pulls it into space?
Gravity acts on all matter, not just water (it just so happens that water flows with less resistance than rock) which is why we get noticeable water tides but not very noticeable earth tides. However, if you were to bring a very large gravitating body too close to earth, you would find that the earth isn't quite as solid as it feels.
The answer to your question is yes, but along with the water that would go into 'space', the earth itself would get ripped to shreds by tidal forces bringing large chunks of earth up into 'space' as well, though the name space doesn't really apply here since it's not more of a very messy debris field or accretion disk.