While I don't think you can get 100% of the solder off, you should be able to get it down to a very thin layer on the pad. You'll want to use solder flux and solder wick to absorb as much as possible off the board. You'll still have a silver color to it, but with enough heat and flux, the majority of the solder should come off.
However, keep in mind that the solder may be a result of an attempted fix rather than the cause of the problem. I'm actually wondering if the U and L directions weren't working before, and the previous owner tried to fix that somehow with solder. Those pads are just electrical contacts with a conductive pad sitting above them. When you press the pad down, it bridges the connection between the two halves of the circle, completing the circuit and activating the direction signal you want.
In order to test the signal, all you should have to do is use a conductive object to short the two sides together. Try that for all four pads and verify that the same operation makes all four directions work. If it does, then you may indeed gain something by cleaning off the solder, but if only the D and R pads work that way then there's something else wrong.
Take a look at doing that and see what happens, then come on back and tell us the result.