Burns was born in Erie, Pennsylvania, and raised in Houston, Texas. In seventh grade, she appeared in a musical production of A Midsummer Night's Dream. In 1970, she made her first film appearance in the Robert Altman movie Brewster McCloud (1970). Burns attended the University of Texas at Austin and graduated from there with a degree in Drama in 1971. Burns was cast in Lovin' Molly (1974), but was replaced by Susan Sarandon. Burns stayed on as a stand-in for Sarandon and Blythe Danner.

In Tobe Hooper's The Texas Chain Saw Massacre (1974), Burns, in her first lead role, portrayed the heroine Sally Hardesty, a teenager who travels with her brother and some friends to the cemetery where her grandfather is buried to investigate reports of grave vandalism, and then encounters an insane, cannibalistic family including the chainsaw-wielding Leatherface. The film was a massive hit, becoming one of the most successful independent films ever at the time.

She also had a small part in George Roy Hill's The Great Waldo Pepper (1975), where she helped cast some of the film's extras.

In 1976, Burns had a role in the television miniseries Helter Skelter about the real-life trial of Charles Manson and his "family". In the series, she played Linda Kasabian, a member of the Manson Family who was granted immunity in exchange for her testimony against the defendants. The miniseries was nominated for three Emmy awards. Recalling her memories of working on Helter Skelter, Burns said: "It was a great experience. But nobody really wanted to touch it [due to the subject matter]. It was like, 'Who wants to be in that picture? Who's actually gonna do that picture?'" Her next role was Faye in the horror film Eaten Alive, which was helmed by The Texas Chain Saw Massacre Director Tobe Hooper. She starred alongside Robert Englund.

Burns subsequently starred in the films Kiss Daddy Goodbye (1981) and Future-Kill (1985). In 1994, Burns briefly reprised her role as Sally Hardesty in the sequel Texas Chainsaw Massacre: The Next Generation (1994). Burns also made a cameo appearance as Verna Carson in the sequel Texas Chainsaw 3D, which was released on January 4, 2013. However aside from these roles and occasional appearances at horror conventions Burns lived a relatively quiet life out of the spotlight in the Houston area during her later years.

Burns, 65, died in her sleep on August 5, 2014 and was found in her Houston, Texas home by her brother Bill; the cause was an apparent heart attack, although not specified.