Shaw was born Philomena 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 liked to be called "Fe Fe" in her childhood years and attended secondary school at Scoil Mhuire In Cork City.

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 Young Woman in Machinal, (1931), 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.[8]

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.

Shaw had a brief but key role in Brian DePalma's The Black Dahlia. She also made an appearance in Sherlock Holmes as a family friend of the widow in The Crooked Man