Burns began his comedy career in 1959, when he partnered with George Carlin; both were working for radio station KXOL in Fort Worth, Texas. After successful performances at a Fort Worth beat coffeehouse, The Cellar, Burns and Carlin headed for California in February 1960 and continued to work together for two more years. An album containing some of their material was released in 1963, titled Burns and Carlin at the Playboy Club Tonight.
Longer lasting was a later teaming with Avery Schreiber, whom he met when they were both members of The Second City, a live comedy and improv troupe based in Chicago. Burns and Schreiber were best known for a series of routines in which Burns played a talkative taxicab passenger, with Schreiber as the driver.
During the first half of the 1965-1966 season of The Andy Griffith Show, in an attempt to replace the Don Knotts' Barney Fife character after Knotts left the show, Burns was cast as Warren Ferguson, a dedicated but inept deputy sheriff. His character was not popular, and was dropped after eleven appearances.
In 1967, he was cast as 'Candy Butcher' in The Night They Raided Minsky's.
Burns voiced Harry Boyle's (Tom Bosley) fanatically reactionary neighbor, Ralph Kane, in the short-lived syndicated primetime cartoon Wait Till Your Father Gets Home. The series was a forerunner of adult animation comedies.
Burns was the head writer for the first season of Hee Haw and The Muppet Show. Schreiber appeared on an episode with The Muppet Show during that first season. Burns also co-wrote The Muppet Movie (with Jerry Juhl, his successor as head writer of The Muppet Show.)
He hosted a 1977 episode of NBC's Saturday Night Live, the first to carry this title (the show was initially called "Saturday Night"), after ABC's Saturday Night Live with Howard Cosell - a different program - was canceled.
In the early 1980s, Burns became a writer, announcer and sometimes-performer on the ABC sketch comedy series Fridays. He and comedian Michael Richards were involved in a staged on-air fight with Andy Kaufman, later re-created in the Kaufman biopic Man on the Moon.
He teamed with Lorenzo Music to provide the voices for a pair of crash test dummies respectively named Vince and Larry in a series of United States Department of Transportation public service announcements that promoted the use of seat belts. Distributed by the Ad Council, the advertising campaign ran from 1985 to 1998. In 1993, Burns starred in the cartoon-series Animaniacs, as the voice of Sid the Squid, giving the character a raspy, Daffy Duck kind of voice. Schreiber also appeared on the show, as Beanie the Bison.
Burns was a guest voice in the a 1999 episode of The Simpsons, "Beyond Blunderdome".