the biggest issues are fuel and weight.
The shuttle gets into earth orbit with pretty much no fuel left on it, so it would require more fuel to accelerate it out of the earth orbit and on it's way to the moon, and that is when the weight issue comes in, the shuttle is pretty heavy so would require a huge amount of fuel to accelerate it towards the moon (and you would need to be carrying enough fuel to get home again remember)
There are a couple of other smaller problems that could probably be overcome, such as the return flight from the moon would cause the shuttle to hit the earths atmosphere at a far higher speed than it was designed to causing much higher temperatures and loads than it's designed to withstand (sure you could use thrusters to slow it down, but that brings us back to the fuel/weight problem again)