Burbeck was born in
Leicester, He played for
England Youth against their Scottish and Welsh counterparts in 1952, and was a member of the squad for that year's
FIFA Under-18 Tournament. He made his Leicester debut in
the Football League on 15 September 1952 in the
Second Division match away to
West Ham United, and played twice more, in the
1955–56 season, before moving on to fellow Second Division club
Middlesbrough in 1956. He made his debut in a 3–1 win against
Port Vale on 6 October, Burbeck remained in the starting eleven for 54 consecutive matches in all competitions, making way in December 1957 for the debut of future
England international Eddie Holliday. He regained his place, only to lose it again after breaking his wrist in January 1959. He returned to the team towards the end of the season, and on 22 October 1960 scored twice in a 6–6 draw at
Charlton Athletic; it was only the second time that such a scoreline had occurred in the Football League. He finished his Middlesbrough career in 1963, with 24 goals from 139 League matches and another 5 goals from 13 cup ties. His league career ended with 18 appearances and a goal in the 1963–64
Third Division season with
Darlington,. He spent 3 seasons at
Edgar Street, making a total of 61 appearances and scoring 18 goals. ==Cricket==