He also inherited the role of Mr. Magoo in the animated segments of the live action feature film of the same name in 1997 (after Jim Backus died in 1989).

Burson was also one of the successors to Mel Blanc, and voiced many of his characters as well, including Bugs Bunny, for whom he was given the responsibility of voicing in 1995's Carrotblanca, a well-received 8-minute Looney Tunes cartoon originally shown in cinemas alongside The Amazing Panda Adventure (USA and Canada) and The Pebble and the Penguin (non-US).

Burson also voiced Bugs in the 1997 short From Hare to Eternity, which is notable for being dedicated to Friz Freleng (who had died in 1995).

Alternating with Jeff Bergman, Bob Bergen, Joe Alaskey, Jim Cummings, Maurice LaMarche, and Billy West, he also voiced several other Looney Tunes characters including Daffy Duck, Porky Pig, Tweety, Sylvester, Elmer Fudd, Yosemite Sam, Marvin the Martian, Tasmanian Devil, Pepé Le Pew, Speedy Gonzales, and Foghorn Leghorn on various Warner Bros. animated television series, films, toys, and video games.

His other voice work includes shows such as CatDog, Batman: The Animated Series, All-New Dennis the Menace, Mother Goose and Grimm, The Angry Beavers, Johnny Bravo, Samurai Jack, The Smurfs, Super Friends, The Twisted Tales of Felix the Cat and Garfield and Friends, the feature film Jurassic Park and the three Star Wars video games The Phantom Menace, Jedi Power Battles, and The Gungan Frontier.