He played junior hockey for the Montreal Junior Canadiens from 1943 to 1946, and was picked up by the Montreal Junior Royals for the 1944-45 Eastern Canada Memorial Cup Playoffs. He scored 41 goals in 40 games in his three seasons for the Junior Canadiens.
Burnett played for the senior Victoriaville Tigres for the 1946-47 and 1947-48 seasons, scoring 76 goals and 146 assists in 100 games over those two seasons. He then played one season for the Sherbrooke Saints.
In 1949 he signed as a free agent with the Springfield Indians of the American Hockey League. He lasted two seasons with the Indians and then relocated with the team to the Syracuse Warriors in 1951. The Warriors loaned him to the Rangers in 1952-53 for his three game NHL career.
He won the league scoring title and the Byng of Vimy Trophy in 1954-55. Burnett continued to play for the Royals through the 1958-59 season in the Quebec League. He played for the team in 1959-60 and 1960-61 in the Eastern Professional Hockey League and retired in 1961 when the team folded.