Burt, who was appointed by Steve Cotterill in December 2020, played an integral part in building a Town squad which finished 12th in League One - the club's second-highest finish in 34 years.
His departure paves the way for a director of football - a role the club approached Cheltenham Town to speak to Mickey Moore about.
Burt was present during Town's 1-0 defeat to Lincoln City on the final day of the League One season - and he was seen embracing the players full-time.
