Dunne was born in County Dublin and raised in County Mayo, Ireland. She trained at the Gaiety School of Acting before beginning her professional career in 2009.She made her acting and writing debut in the television film Dublin Stories and appeared in two episodes of Rental Boys. In 2010, she co-wrote and appeared in The Taste of Home. In 2014, she wrote and starred in the RTÉ One comedy The Walshes as Carmel Walsh.

Since 2016, Dunne has portrayed Anne Flynn in the BBC Two comedy Motherland, appearing in all 19 episodes. In 2019, she began appearing in the Channel 4 sitcom Derry Girls as Geraldine Devlin, the mother of established character Clare Devlin. Dunne appeared in the series on a recurring basis until its end in May 2022. Dunne was then cast in the BBC One medical comedy-drama series This Is Going to Hurt as Ria, the receptionist. The series began airing on 8 February 2022. She also appeared as a contestant on Celebrity Mastermind, with her specialist subject being the 1980 Stanley Kubrick film The Shining.

Dunne is married; she and her husband have one child, born in late 2020. Dunne's infant child appeared in the third series of Motherland as Niamh, the daughter of Dunne's character.