He was discovered in the 1950s by Harry Dillon, who ran the 37 Theatre Club on the top floor of his shop the Swiss GEM Company, 51 Lower O'Connell Street Dublin.
O'Shea began acting on the stage, then moved into film in the 1960s. He became popular in the United Kingdom, as a result of starring in the BBC sitcom Me Mammy alongside Yootha Joyce. In 1967–68 he appeared in the drama Staircase, co-starring Eli Wallach and directed by Barry Morse, which stands as Broadway's first depiction of homosexual men in a serious light. For his role in that drama, he was nominated for a Tony Award in 1968.
O'Shea starred as Leopold Bloom in Joseph Strick's 1967 film version of Ulysses. Among his other memorable film roles in the 1960s were the well-intentioned Friar Laurence in Franco Zeffirelli's Romeo and Juliet and the villainous Dr. Durand Durand (who tried to kill Jane Fonda's character by making her literally die of pleasure) in Roger Vadim's counterculture classic Barbarella (both films were released in 1968). In 1984, O'Shea reprised his role as Dr. Durand Durand (credited as Dr. Duran Duran) for the Duran Duran concert film Arena, since his character inspired the band's name. He played Inspector Boot in the 1973 Vincent Price horror/comedy film Theatre of Blood.
O'Shea's first wife was Maureen Toal, an Irish Actress, with whom he had two sons. He divorced her in 1974 and later married Kitty Sullivan, whom he met in Italy, where he was filming Barbarella and she auditioning for Man of La Mancha.
Other notable stage appearances include Mass Appeal (1981) in which he originated the role of "Father Tim Farley" (for which he was nominated for a Tony Award as "Best Actor" in 1982), the musical Dear World in which he played the Sewer Man opposite Angela Lansbury as Countess Aurelia, Corpse! (1986) and a 1994 Broadway revival of Philadelphia, Here I Come.
He was active in American films and television, such as his memorable supporting role as the trial judge in the Sidney Lumet-directed movie The Verdict (1982) with Paul Newman, an episode of The Golden Girls in 1987, and portraying Chief Justice of the United States Roy Ashland in the television series The West Wing. In 1992, O'Shea guest starred in the season 10 finale of the sitcom Cheers, and, in 1995, in an episode of the show's spin-off Frasier. In the episode of Frasier, he played Dr. Schachter, a couples therapist who counsels the Crane brothers together. He appeared in the pilot episode of Early Edition as Sherman.
O'Shea received an honorary degree from Quinnipiac University in 2010.
O'Shea died on 2 April 2013, in New York City following a short illness at the age of 86.