His cause of death is not known but his sister, Loretta Kelly, said he had regularly been in hospital over the past year.
She added he had not received a diagnosis in that time but his health was so poor he had not been able to see recent spin-off film Creed.
Burton played Tony “Duke” Evers, who trained both Rocky and Apollo Creed.
The character supported Creed, Rocky Balboa’s opponent, in the first two boxing films before later becoming the trainer to Balboa himself.
Burton’s co-star Carl Weathers tweeted: “Sad news. RIP Tony Burton. His intensity and talent helped make the Rocky movies successful.”
His former classmate Gerald Moore said: ”It’s just like someone in my own family passed. Whenever we got together we would reminisce on old times.”
Before becoming an actor, Burton – who was from Flint, Michigan – briefly fought professionally and won Flint Golden Gloves light heavyweight titles in 1955 and again in 1957.
Burton is seen briefly in Creed, which was released in the UK last month, but only as part of a flashback which used stock footage from an earlier film.
In addition to appearing in the Rocky Films, Burton also appeared in The Shining and Stir Crazy.
His last role was Sheriff Stoker in comedy-horror Hack! released in 2007.
According to IMDb, he had signed up to appear in The Truth About the War in Heaven: Declaration of War, but the film has not yet gone into production.
Burton’s TV work included appearances in The A-Team, NYPD Blue, Kojak and Twin Peaks.