Mulligan was born on November 13, 1932, in New York City. After attending Columbia University, Mulligan began working in theater, making his debut as a stage manager and performer on Broadway in All the Way Home in 1960. Additional theatre credits included A Thousand Clowns, Never Too Late, Hogan's Goat, and Thieves.

Mulligan made a brief, uncredited appearance in the 1963 film Love with the Proper Stranger, which was directed by his elder brother. He starred with Mariette Hartley in the 1966-67 season comedy series The Hero, in which he played TV star Sam Garret. Garrett had starred in a fictional series as Jed Clayton, U.S. Marshal. The Hero lasted for 16 episodes. Another notable TV appearance was on the I Dream of Jeannie episode "Around the World in 80 Blinks", as a navy commander accompanying Major Nelson (Larry Hagman) on a mission.

Mulligan married four times. He was first married to Patricia Jones from 1955 to 1960, with whom he had a son, James. That was followed by marriages to Joan Hackett from January 3, 1966, to June 1973 and Lenore Stevens from 1978 to 1990. His last marriage was to adult film Actress Rachel Ryan on April 27, 1992, which only lasted to early 1993.

Mulligan's most notable film role was as General Custer in Little Big Man, whom he portrayed as a borderline psychotic. He also appeared in the disaster movie spoof, The Big Bus (1976), where he was reunited with Larry Hagman, with whom he starred in an episode of I Dream of Jeannie (see above). As for his radio work, he starred in the adaptation of Edgar Allan Poe's "The Oblong Box" heard on The CBS Radio Mystery Theater (1975).

His highest profile roles in television were as Burt Campbell in the sitcom Soap (1977–81), for which he won a Best Actor Emmy Award, and as Dr. Harry Weston in the NBC series Empty Nest, a spinoff of The Golden Girls in which his character had appeared in a couple of episodes. Empty Nest ran for seven seasons, and Mulligan won a Best Actor Emmy Award as well as a Golden Globe Award for his performance. He also played Secretary of State william Seward in Lincoln (1988), a TV movie based on Gore Vidal's novel.

Mulligan returned to perform on Broadway and in films, in which he usually played supporting roles. A notable exception was the black comedy S.O.B. (1981), in which he played lead character Felix Farmer, a Hollywood producer-director modeled on the film's actual producer-director, Blake Edwards. The film again featured Larry Hagman. Mulligan was cast as Reggie Potter in the television series Reggie (1983). Lasting for only six episodes, it is a loose adaptation of the BBC series The Fall and Rise of Reginald Perrin. In the 1984 film Teachers, he played an eccentric high school history Teacher (who in one scene teaches the Battle of the Little Bighorn, once more dressed as General Custer).

In 1985, he appeared in The Twilight Zone episode "Night of the Meek", where he took on the role of Henry Corwin, an alcoholic department store Santa Claus who becomes the genuine article, in the remake of the 1959 Christmas episode "The Night of the Meek", the character that Art Carney had played in the original version. The next year, he appeared in another episode of the series, "The Toys of Caliban".

He appeared in Disney's 1988 film, Oliver & Company, as the voice of the oafish Great Dane named Einstein. His final performance was a voice over on Hey Arnold! in 2000 as the voice of Jimmy Kafka, the long mentioned but never seen former friend of Arnold's Grandpa.

On September 30, 1993, Mulligan received a star on the Hollywood Walk of Fame for his work in the television industry, located at 6777 Hollywood Blvd.

On September 26, 2000, Mulligan died of colorectal cancer at his home in Los Angeles, California. At his own request, his remains were cremated and there was no funeral Service. Mulligan was survived by his son James Mulligan from his first marriage, and two brothers, Robert and James.