The Moon collapses on itself - cheese can't handle the pressure you'd be getting past the first few hundred feet of depth. When the dust settles, the Moon is a ball of hyper-compressed organic matter (with an outer layer of, well, cheese, which gets utterly destroyed by the extreme temperatures of the Moon's day-and-night cycle). It thus ends up occupying a significantly smaller volume than before...and thus, gives us less moonlight. I suspect this results in a very slight temperature drop on Earth.
More importantly, with the Moon's mass and gravitational pull being reduced by an order of magnitude, tides almost disappear. This fucks up the environment something fierce.