Burns was born in Glasgow, and started his career with Rangers as an apprentice but left for Birmingham City aged 17, in 1971. He arrived at the Blues as a defender, but was converted to striker after Bob Latchford left in 1974, and earned the first of his 20 international caps in that role soon afterwards. However, after joining Nottingham Forest for £150,000 in 1977, he was converted back into a central defender by Brian Clough and Peter Taylor.
Burns proved a great success in his new role and was voted FWA Footballer of the Year in 1977–78, as Forest won the First Division title. He was an influential figure in the side's European Cup campaigns over the next three seasons, his defensive partnership with Larry Lloyd instrumental to Forest's victory in the 1979 and 1980 tournaments. He also managed to score in the second leg of the 1979 UEFA Super Cup final victory against Barcelona. At one time he was renowned for his 'wild man' image and while maintaining an aggressive nature, he also possessed a shrewd football brain and was always likely to score vital goals.