Pity the poor Unicorn. Written, I'm told, by Shel Silverstein and made famous by the Irish Rovers (who sang it on Ed Sullivan or some such show). That's the only reason it's become an "Irish" song.
Another Shel Silverstein song that goes over well on St. Patrick's Day is The Mermaid Song. Lively and suggestive, just they way audiences tend to like 'em.
We usally do these songs, which have gone over well: The Orange and The Green, We're All Off to Dublin in the Green, Jug of Punch, Galway Bay (one of those Tin Pan Alley Irish-American numbers), Ballybae, Haul Away Joe (any rousing chantey is likely to be a hit).
Perennial requests include: Four Green Fields, When Irish Eyes are Smiling (which is more interesting if you sing the verses and not just the chorus), I'll Take You Home Again, Kathleen, Wild Colonial Boy.
Have fun, and bring lots of extra guitar strings.