Legal Issues
A number of questions come up whenever you mention dual citizenship to anyone. "Will you still be a US citizen?" "Do you have to pay taxes for both countries?" "Will they break your knees when you try to come back to the US?" "Could you be drafted by both countries?" etc.
If taking the citizenship oath of a country involves renouncing former allegiances, you're in trouble. You will be legally forfeiting your US citizenship. Don't do this unless you know what you're getting into.
Fortunately, Ireland doesn't have this provision in their oath of allegiance. Basically dual citizenship means that you fall under the laws of both countries. The US takes a "don't ask" stance on dual citizenship. It doesn't matter what other citizenships you have, as long as they don't interfere with US law.
You pay taxes as a US citizen, even when you are living overseas (as US expats do). You don't have to pay taxes to Ireland as an Irish citizen living overseas, but when you are living in Ireland you do pay taxes. If both countries had a draft, you could be drafted by both countries, and you would be expected to appear for both nations. Fortunately neither Ireland nor the US have a draft as a current institution.
There's a very good website here on dual citizenship.