I want to install a window A/C unit in the wall of my third floor. I know that if I vent it into the attic, it will increase the temperature of the attic. What problems would occur if the heat in the attic did increase significantly?

Actually the A/C will not work well at all. They depend upon ground level temperatures to cool the condenser/to exchange heat.
Now when you put in a hot attic the unit will have a impossible time disposing of the heat. Not recommended at all. Just when you need the A/c the most the attic air will be way to hot to remove the heat from the unit.

hvac1000 is correct and it most likely will violate local and state fire codes. Most apartments use the sheetrock as a fire rated barrier and cutting any holes would be a violation. Not a good idea...