Mickey, I hardly know where to start but a short list of songs that aren't Irish, but are requested as such are:
Three Score and Ten
Rose of Allandale
The Grey Funnel Line
Aragon Mill
Danny Boy (most requested song of all)Of course the tune may be Irish.
How Can I Keep from Singing
Freeborn Man (and many of Ewan McColl's other songs)
Willie McBride
Will Ye go Lassie Go
The Water is Wide
Hard Times Come Again no More
The Great Silkie of Shule Skerrie
Amazing Grace
Also anything, absolutely anything, that Mary Black has recorded.

As for genuine Irish songs I actually said that if I sing one
it's not what the requester wants - because they really want one of the above. If I talk people into an unfamiliar song they don't dispute my claims of it's origins if I say it's Irish, because they believe that most songs are Irish anyway!
Some songs that I sing that are Irish or of Irish origin, as far as I can find out are:
Shule Agra/Buttermilk Hill (in America)
Blackwater Side (may be English but tune sounds Irish)
Dublin City, Tell me Ma, and several others from the Clancys that they said were Irish.
As I don't understand or speak Irish Gaelic (although I have some Irish roots) I don't sing any songs in Irish.
How are you Mickey? Nice to meet again.