You can damage the capacitor by shorting it, because they are rated to release so much current at a time - but NOT a short. I'm assuming you are replacing aluminum electrolytic or tantalum caps, though. These need to be soldered in correctly (they are polarized). Make sure you note the original caps' orientations, otherwise you may damage much more of the board.
To ensure the caps are discharged, simply remove the board from power for a long time, like a day or so. The internal resistance will slowly discharge the caps, as well as any resistance across the caps.
Like Brett said, make use of insulation.
If you can, reduce risk of ESD by grounding your skin with a ESD wrist strap.