Dunn was born in Chicago, Illinois, the daughter of Margaret (née East), a nurse, and John Dunn, a musician and poet. Her brother is actor Kevin Dunn. She was raised in a Catholic family, and has Irish, English, Scottish, and German ancestry. Dunn attended the School of the Art Institute of Chicago from 1970 to 1972, San Francisco City College, and the Academy of Fine Arts.
Dunn joined SNL in 1985 with the return of Lorne Michaels as executive producer. The 1985–1986 season proved to be a ratings disaster, and she was one of only five cast members who were not fired at its end (the others were newcomers Jon Lovitz, A. Whitney Brown, Dennis Miller and longtime featured player Al Franken). Dunn's characters included half of "The Sweeney Sisters" lounge act alongside Jan Hooks, as well as talk show host Pat Stevens (which became a popular recurring role starting in the low-rated 1985–1986 season), melodramatic French hooker Babette, and film buff Ashley Ashley from the "Actors on Film" sketch from the 1985–1986 season. Her impressions included Ann Landers, Imelda Marcos, Liza Minnelli, Tyne Daly, Joan Baez, Martina Navratilova, and Cokie Roberts. Dunn made headlines in 1990 when she, along with original musical guest Sinéad O'Connor, boycotted an episode which was hosted by comedian Andrew Dice Clay because they found his misogynistic humor offensive. SNL alumnus Jon Lovitz discussed Dunn's boycott of the show in detail during an episode of "The ABC’s of SNL" with director Kevin Smith. "It's the last show of the season. Nora...she caused a lot of trouble. [She] was very hard to get along with, and they weren't going to ask her back, anyway...And this is how the press works. And I'm telling you I'm on the inside of this. They don't know the story. It's like she's just doing it to get press. It's her last hurrah. They're not asking her back onto the show."
Dunn appeared in recurring roles on Sisters from 1993–1996 and The Nanny from 1998–1999, as well as a guest-starring role in a two-part episode of The X-Files in 1998. She appeared on LOL in 2012. In the 8th episode of the 9th season of crime procedural drama Bones, she played author Tess Brown, a feuding rival of protagonist Temperance Brennan. In 2015 she appeared in the third season of New Girl. Her film work includes Working Girl (1988), How I Got into College (1989), Miami Blues (1990), I Love Trouble (1994), Shake, Rattle and Rock! (1994), Three Kings (1999), Zoolander (2001), Max Keeble's Big Move (2001), Bruce Almighty (2003), Runaway Jury (2003), Love for Rent (2005), Pineapple Express (2008), My Suicide (2009) and Entourage (2015). In 2014, Dunn began appearing in a series of commercials for Clorox. She has also been a storytelling contributor to Chicago's Under the Gun Theater.