[1] O'Shea was born and brought up in Dublin and educated by the Christian Brothers at Synge Street school,[2] along with his friend Donal Donnelly.

[3] 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.

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.

[1] 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 die of pleasure) in Roger Vadim's counterculture classic Barbarella (both films were released in 1968).

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.

He was married to the Irish actress Kitty Sullivan, whom he met in Italy, where he was filming Barbarella and she was auditioning for Man of La Mancha.