Kaye played reluctant resister and cafe owner Rene Artois in the show, which was set in Nazi-occupied France during the Second World War.
The star’s agent confirmed he died on Monday but made no further comment.
The show ran on BBC1 from 1982-1991 and remains one of Britain’s best-loved sitcoms.
As Rene, Kaye risked his neck to aid the resistance and stay out of trouble with the Nazis, with Carmen Silvera as his wife Edith, Richard Gibson as Gestapo officer Herr Flick and Officer Crabtree (Arthur Bostrom), whose mangled vowels led him to adopt the greeting: “Good moaning.”
A long-running joke had Rene hiding a painting, The Fallen Madonna With The Big Boobies, inside a garlic sausage.
He also starred in Coronation Street, playing Elsie Tanner’s nephew Bernard Butler on the soap from 1969 to 1970.
Kaye’s last acting credit came in the BBC sketch show Revolver in 2004.