Shaw was born Fiona Mary Wilson in County Cork, Republic of Ireland to a mixed-religious couple and was raised Catholic. Her father was an eye surgeon and her mother was a physicist. She received her degree in University College Cork. After training at the Royal Academy of Dramatic Art (RADA) in London, she received much acclaim as Julia in the National Theatre production of Richard Sheridan's The Rivals (1983), a role which demonstrated her gift for comedy. Despite her natural comic abilities, Shaw has opted more often than not for roles showcasing her extreme but unaffected emotional intensity. These performances have earned her numerous stage awards.

Her notable theatrical roles include Celia in As You Like It (1984), Madame de Volanges in Les Liaisons Dangereuses (1985), Katherine in The Taming of the Shrew (1987), Winnie in Happy Days (2007), and the title roles in Electra (1988), The Good Person of Sechuan (1989), Hedda Gabler (1991), The Prime of Miss Jean Brodie (1998) and Medea (2000). She performed T.S. Eliot's poem The Waste Land as a one-person show at the Liberty Theatre in New York to great acclaim in 1996.

She controversially played the lead in Richard II, directed by Deborah Warner in 1995. Shaw has collaborated with Warner on a number of occasions, on both stage and screen. Shaw has also worked in film and television, including My Left Foot, Jane Eyre, Persuasion, Gormenghast, and a number of the Harry Potter films in which she plays the insufferable Aunt Petunia.

For a number of years, Shaw has been romantically linked in the press with actress Saffron Burrows.[10][11][12] Neither actress has publicly confirmed or denied the relationship. The two appeared together in the National Theatre's production of The PowerBook,[13] a play based on the novel of the same name by Jeanette Winterson in which they played lovers.