First let me start of to tell you that only an emotionally or intellectually extremely disturbed or even mentally retarded person would even consider renouncing their US citizenship. There is no advantage to this at all until you come to a point where you have an income in the 7 digits or higher, but if you make that kind of money, shouldn't you run through life with a smile on your face? If you could sell your US citizenship, and oil sheik would easily give you $1 million for it, without blinking an eye.
Now to your question: the US and Ireland have a bilateral treaty. If you pay taxes on your income in the U.S., Ireland won't tax you on it, and if you were not a citizen of the U.S. anymore, then you'd have to pay income taxes in Ireland on this, which is higher than in the United States. So even if you have an income of . . . let's say $1,000,000 per annum, then you either pay Uncle Sam or you pay the Irish government their share.
If you want tax free, seek residency in Monaco, the Cayman Islands, or Luxembourg. Once you make about a million a year, that's the place to be.
An immigrant from Europe, I live in the charming old mission town San Buenaventura and work as an attorney in Santa Barbara, California.