The Cliffs of Moher truly feel like the edge of the world. They rise almost 700 feet high straight out of the Atlantic, and the cliffside is not roped off so you can (very carefully) walk almost to the edge. There's a path to O'Brien's Tower which offers lovely views. If you're anywhere near County Clare, make a stop at the Cliffs, which must be seen to be really appreciated.
The foot & mouth restrictions do limit what you can see in Ireland (outside the cities) at the moment. Check the official Ireland tourist website for details on what is open.