Burns was a theatrical late bloomer, as he "experienced life" for two years after high school before going to college and it took some time in college before he was drawn to acting. After earning his degree, he moved to New York where he began appearing in Off-Broadway productions, including a version of "Don Juan" for the New York Shakespeare Festival. He also made his TV debut in a small role in the 1984 ABC Afterschool Special "Mom's on Strike". Within three weeks of moving to L.A. in 1985, he was cast as a pathological rapist in several episodes of the NBC drama series "Hill Street Blues". In 1987, he landed the recurring role of a body collector on "Max Headroom" (ABC). The next year, Burns was cast in the role of Kirk Morris, the member of the club for divorced adults least likely to win the heart of a judge, in "Dear John" (NBC, 1988-92). In 1993, he was slotted into "Bob" (CBS), the least successful Bob Newhart series, as an obnoxious sales vice president. The following season, Burns was added to the cast of NBC's "The Mommies", as a sarcastic, homebody father named Tom. He was next cast as Mel Harris' husband, a teacher, working to make their marriage and blended family succeed in the sitcom "Something So Right" (NBC, 1996-97; ABC, 1997-98).
Burns' work in TV-movies and films has been sporadic. He played the villainous Cade Dalton in the NBC miniseries "Luck of the Draw: The Gambler Returns" (1991). He made his feature film debut as a hockey player in the 1986 Michael Keaton vehicle "Touch and Go" and also appeared in "Wired" (1989) and as one of Kirk Douglas' relatives in "Greedy" (1994).