After a short stint in the United States Marine Corps during his late teens, Carr launched his acting career with a role in a New Orleans production of Herman Melville's Billy Budd. By the middle 1950s, he was working on live television in New York City, including appearances on the popular Studio One and Kraft Television Theater, while continuing theatrical work in stock companies in Ohio and Michigan, including roles such as Peter Quilpe in The Cocktail Party, Haemon in Antigone, Jack in Tennessee Williams' The Rose Tattoo, and Hal Carter in William Inge's Picnic. He toured in summer stock with Chico Marx in Fifth Season.
Carr made his film debut in 1955 with a small uncredited role in Alfred Hitchcock's thriller The Wrong Man. That same year, he portrayed a prisoner of war in the New York Theatre Guild production of Time Limit on Broadway. His film career continued with a much larger role in Alfred Werker's The Young Don't Cry in 1957 starring James Whitmore and Sal Mineo; and that same year he appeared in the Warner Bros. rock and roll jukebox movie Jamboree as Pete Porter.
He worked steadily on television in the late 1950s and early 1960s with guest spots and supporting roles in many western series such as three appearances on Laramie, Trackdown, four appearances on Rawhide, The Rifleman, The Tall Man, The Travels of Jaimie McPheeters, and The Virginian. He also appeared in many dramas. One such appearance was in 1964 when he played folk singer and defendant Con Bolton in the Perry Mason episode "The Case of the Tandem Target". He also appeared on 77 Sunset Strip, Straightaway, The Everglades, Dr. Kildare, Going My Way, Hawaii Five-O, The Fugitive, Twelve O'Clock High, and The Silent Force, interspersed with occasional film work, including Captain Newman, M.D.. Other television appearances were on Burke's Law, Combat!, Gunsmoke, The Time Tunnel, and The Invaders.
In 1965, Carr won the role of Bill Horton, the physician son of protagonist Dr. Tom Horton on Days of Our Lives in its first season. He was later a regular on General Hospital and The Doctors. Carr went on to work in dozens of other television shows in the intervening years, including Get Smart, Mannix, The Rockford Files, and Murphy Brown. He may be remembered best, however, for his various appearances on science fiction shows over the years. In 1964/1965, he had the recurring role of uptight crewman Casey Clark on Voyage to the Bottom of the Sea.
In 1966, Carr played Lt. Lee Kelso, the affable USS Enterprise helmsman who is strangled psychokinetically by the ship's rapidly mutating navigator in the second Star Trek pilot episode, "Where No Man Has Gone Before", giving him the dubious honor of being the first dead Red Shirt in Star Trek history, although nine off-screen deaths had previously been mentioned in the same episode. In 1981, he joined the cast of Buck Rogers in the 25th Century as 'Lt. Devlin', one of the officers on the Earth Starship Searcher.
Throughout his career, Carr's first love was the stage. He appeared in nearly 100 stage productions on Broadway, off-Broadway and off-off-Broadway, as well as touring companies, stock, and in regional theaters around the United States. He received the LA Weekly Theater Award for Best Actor in the Theatre East production of Manhattan Express in 1987 and garnered a 1995 Dramalogue Award for his role in the Los Angeles Repertory production of Assassins. Carr was also a writer and director, and headed the Play Committee of the L.A. Repertory Company.