Well actually I saw a photo of one of those signs in a history book about the Irish in America. It was a advert for some kind of job and the last line said No Irish Need Apply.
Here's a learned article on the myth of "NINA" in America:
"No Irish Need Apply" - a myth of victimization
My daughter took a history course on the Irish in America, and that's what her prof. said (she attends a conservative college with a strong academic program). I got curious and started investigating myself, and while you can find all sorts of mock-ups from the music hall, movies, and so forth, you can't actually find a contemporary example that was actually posted in somebody's window.
The article explains the origin of the story, and why it was popularized.