The "No Irish Need Apply" myth was comprehensively demolished by a scholar who set out to quantify its extent, not disprove it. But what he discovered in his research is that it didn't happen.
Well, there is this from the New York Times in 1854:
I does seem to be the only one on the Internet, though.
I noticed that people always put the time the signs appeared roughly in their grandparents time. So you'd have politicians in recent years talking about "No Irish Need Apply" signs in Boston in the 1930s, a time when the Irish largely ran the city, at least politically, and such signs weren't in evidence.
So far as I can tell the current version is that at one point in the early 19th century such signs or lines in ads appeared London, England. Irishmen wrote songs about that and when the songs were passed down the legend grew that such signs were common in the US.
Something similar may be true of the "No Dogs or Jews" signs people claim to have seen on the lawns of fancy hotels. There may well have been such signs in Nazi Germany, but it's doubtful to me that any upscale hotel would be so blatant or that that particular wording was common in the US. There were other ways of getting the message across (i.e. restricted clientele).
Truly excellent article by the guy who demolished this myth without really planning to.
http://tigger.uic.edu/~rjensen/no-irish.htm
In the entire file of the New York Times from 1851 to 1923, there are two NINA ads for men, one of which is for a teenager. Computer searches of classified help wanted ads in the daily editions of other online newspapers before 1923 such as the Booklyn Eagle, the Washington Post and the Chicago Tribune show that NINA ads for men were extremely rare--fewer than two per decade.
Unlike the employment market for men, the market for female servants included a small submarket in which religion or ethnicity was specified. Thus newspaper ads for nannies, cooks, maids, nurses and companions sometimes specified "Protestant Only." "I can't imagine, Carrie, why you object so strongly to a Roman Catholic," protests the husband in an 1854 short story. "Why, Edward, they are so ignorant, filthy, and superstitious. It would never do to trust the children alone with one, for there is no telling what they might learn." Intimate household relationships were delicate matters for some families, but the great majority of maids in large cities were Irish women, so the submarket that refused to hire them could not have been more than ten percent.