An Irish linguist can chime in, but I wouldn't expect there to be large-scale borrowing into the lexis here—the contact situation of French to Irish was different to that of English. The Normans had a very firm grip on power after the Conquest in England, and French was ascendant in terms social prestige—it was firmly the language of power and politicsl. But this is not such a clear cut case in Ireland. Anglo-Norman settlers in Ireland did not possess the same kind of prestige—in fact, the reverse was true!—because the Irish upper classes were far from wiped out. The Anglo-Normans in Ireland, in fact, were gaelicised on a large scale, adopting Irish culture and customs; this shows that Norman culture was not able to exert a large amount of social dominance over the Irish. Without this, opportunities and motives to borrow become more limited.

What is your source that there are no influences in Irish from French?

Irish has certainly been influenced by English, so you could say it was influenced by French by proxy.

Is there somewhere I can read more about the influx of French words into English?