Well, if we are talking about written evidence of any tunes of Irish authorship (such as religious music), rather than strictly traditional folk tunes, then yes, there are tunes which can be dated back in the manuscripts.

However, I will state again, I know of no Irish traditional music scholars, and there are a number of them nowadays, who will claim there is authorship of traditional tunes which can be verified as being older than a few hundred years, and I'm not aware of any which predate the Middle Ages.

But this argument is really a bit silly. How old a tune is tells you nothing about the music. It just tells you it might be an old tune.

The Irish Traditional Music Archive in Dublin, on-line here:

http://www.itma.ie/home/itmae1.htm

gives a bit of the history. Combined with the above book suggestions Guest ll, you've got some work ahead of you if you have a serious interest in Irish music history.

There are a lot more academic resources, particularly about music from the Middle Ages, but it doesn't seem to be what you are looking for, so I'll leave it at that.