Burr scored 50 goals in Junior B with the local Steeplejacks before joining the OHL's Kitchener Rangers in 1983-84. He scored 41 goals in his rookie year and was chosen seventh overall by the Red Wings at the 1984 NHL draft. The club gave him a look early in the 1984-85 season before sending him back to junior. Burr was dominant in 1985-86 with 60 goals then joined the NHL on a full-time basis the next year.
The young forward scored 22 goals as a rookie in 1986-87 and played solid defensive hockey under coach Jacques Demers. He also demonstrated a willingness to take the body and helped the Wings reach the semi-finals in 1987 and 1988. In 1989-90, the rugged forward scored a personal best 24 goals and represented Canada at the World Championships. He continued to go up and down his wing and see time on both specialty teams through the 1994-95 season. During his last year in Detroit, Burr helped the club reach the Stanley Cup final for the first time since 1966.
Burr was acquired by the Tampa Bay Lightning prior to the 1995-96 season. He was a model of sound defensive hockey for the younger players and chipped in with 27 goals over two years. He joined the improving San Jose Sharks in an off-season deal in June 1997 and was a solid role player that year. He spent much of the 1998-99 schedule with the Kentucky Thoroughblades of the AHL and played a few games for the Tampa Bay Lightning the next year. Burr retired after spending most of the 1999-2000 season in the IHL with the Detroit Vipers and Manitoba Moose.
Courtesy of the Hockey Hall of Fame