The phrase turned up 29 times in the New York Times on November 10, 1854. A variation, Irish Need Not Apply" appeared 7 times. Other ads specified interest in Americans or Protestants, appearing several times on May 1, 1855, which effectively eliminated Irish Catholics.
A song "No Irish Need Apply", written by Kathleen O'Neil, was inspired by a young Irish woman searching for work as a maid in London. She spots a sign in a window which reads: "A small active girl to do the general housework of a large family, one who can cook, clean and get up fine linen, preferred. No Irish Need Apply."
(The London Times, February, 1862)
Nineteenth Century British writer Anthony Trollope explained the pervading sentiment at the time:
"Often depicted as monstrous beings or apes in satirical cartoons, the Irish were not seen as welcome members to English society. Irish immigrants were seen as lazy, drunk, anarchistic criminals whose sole purpose in life was to steal the jobs of English workers. It comes as no surprise, then, that English employers were not very welcoming of Irish workers."(https://apps.cndls.georgetown.edu/projects/borders/items/show/86)
The city of Boston, Massachusetts was a common destination for the Irish. In one year, the Eastern Seaboard city swelled from 30,000 to 100,000 Irish. Boston shop windows often displayed the trademark "No Irish Need apply" signs, relegating the Irish to the most menial jobs. In fact, in the mid-1800's, 70% of Boston's servants were Irish. But they got their foot in the door.
The influence of Irish Catholics slowly grew when the Irish accepted jobs in the police force and in politics. John Francis "Honey Fitz" Fitzgerald, father of the future Rose Kennedy, became mayor of Boston in 1906. Joseph P. Kennedy rose to be the American Ambassador to Britain during the Second world War. His son, John F. Kennedy, of course, became the 35th President of the United States in 1961, the first Catholic to be elected to the position.
Today, No Irish Need Apply signs are proudly mounted in the suburban Boston homes of third, fourth and fifth generation Irish. (http://www.bostonmagazine.com/2006/05/no-irish-need-apply/)
NINA sign circa 1916 courtesy Fulton Street Sign Co.