Sea levels are rising due to global warming. The Moon brings tides that also rise the sea level. If we destroyed the Moon could this problem be solved? Would it have other consequences?
The moon is amazing in the sense that it not only helps accentuate the tides and provide something for home-astronomers to rabbit on about during a first date, but also that it acts as a giant celestial counter balance to Earth's orbital wonkiness.
Without it, our Earth's irregular axis of rotation will cause the planet to "wobble" (kilter back and forward along the lowest point of it's axis) causing ever worsening climate disturbances, such as super-cell storms, tsunamis, earthquakes, catastrophic destruction of eco-systems and the eventually destabilisation of our orbit.
In saying this, removal of the moon would have a visible effect on the oceans tides but would not prevent sea-level fluctuations all together - primarily due to the Sun as explained further in other answers. Hence, we should not be trying to obliterate our closest celestial body in consequence of our industrial practices, but curve greenhouse emissions caused by such practices and from anything else that constitutes as "bad" for the environment now-days.
Therefor, in absence of the moon, yes, tides would be altered somewhat, but in the long run, an increased sea-level will be the least of our concerns... So for the prolonged existence of modern life, let's all avoid destroying the moon at all costs (at least until we run out of fossil fuels - but thats a topic for another day).
This question does not take into account the Sun effect on the tides, which is in fact greater than the one made by the moon. So this answer explains why removing tides does NOT curb the oceans rising problem
I don't think so, because, you are right, when the moon is "above" the ocean, the water is rising. But this additional quantity of water does not come from nowhere, so when the sea level is rising somewhere, it is going down somewhere else.
The thing with the ice melting is that you add water to the overall quantity by releasing the water contained in the ice on the land. So by destroying the moon (and the tide effect of the Sun, too) you would just make an average sea level which would be rising over time.
Tide rise sea level during high tide by lower it during low tide so destroying the moon will just smooth sea level variation.
It can be useful for zone flooded during hight tide: for theses zone it will be like the sea level decreased, but it will change nothing for zone under the medium sea level.
At last, the loss of moon will have heavy consequences on planets balance and could even take earth out of his orbit.
Apart from having devastating ecological effects, especially on marine life were large parts of the biome have their breeding cycles governed by tidal effects. (Not to mention the ecology of the inter-tidal zones around the globe which are also important food generating areas for humanity.) You would therefore be doing more harm than good for the world, humanity included. So there is a key point you need to understand.
Tides don't increase water levels around the planet instead what they do is redistribute whatever preexisting volume of water there is.
Take a glass - mark a line on it and fill it with water to that level. Now slowly tilt the glass from one side to the other. As you do this you will see the water level on one side of the glass rise above the line and drop below it on the opposite side. This is analogis to the effect the moons gravitational pull has on the Earths Oceans as it orbits.
Now take that glass and put in under a dripping tap. The water level will rise and continue to do so until you turn it off. Again tilting the glass slightly makes no difference one way of the other.
The Earths ocean levels are rising because global warming is causing the vast reserves of fresh water trapped on land in the Antarctic, Greenland and elsewhere to melt. This slowly but steadily adds to the volume of water in the oceans. Or if you wish humanity has left the tap dripping and until we bother to turn it off the oceans are going to keep rising in volume - regardless of whether the moon is there or not.
I'm not sure about the change in Earth's orbit thing, but the accepted answer states that the sun has greater effect on tides than the moon. According to this thread that is a misconception.
The tidal forces decay with the cube of the distance between bodies. So, even the sun being much more massive than the moon, it is also much farther away. Thus the moon effect on tides is stronger.