Burn began his career with the Sunderland academy at the age of 13 before joining Middlesbrough in March 2013.[7] He signed a professional contract with the club in May 2014.[8] During the 2014/2015 season Burn captained the Middlesbrough Under 21 team to win the treble, they won the Under 21 Premier League, Final Third Development Cup and The North Riding Senior Cup[9]
On 3 August 2015, Burn joined League One club Oldham Athletic on loan until 1 January 2016.[10] He made his first-team debut on 8 August 2015, starting in a 1–1 draw away to Walsall.[11] He scored his first goal on 19 September 2015, with a header in the 34th minute of a 1–1 draw away to Doncaster Rovers.[11][12]
Burn signed for League One club Bristol Rovers on 26 January 2017.[15] He made his debut on 18 February 2017 in a 1–1 draw away to Port Vale, coming on as a half-time substitute for Tom Lockyer.[16] He made his first start in a 1–0 defeat to Coventry City on 25 March 2017, and his performance drew praise from the Bristol Post, describing him as a "more than able deputy for the absent [Ryan] Sweeney".[17]
On 28 November 2017, Burn joined National League North club York City on loan until 6 January 2018.[18] He made his debut on 2 December 2017, starting in a 2–1 away defeat to Boston United.[19] Burn scored his first goal for York on 26 December 2017 with a 12-yard shot in a 4–1 away victory over North Ferriby United.[20] Having made five appearances and scored one goal, his loan was extended to the end of the 2017–18 season.[21] He finished the loan with 21 appearances and three goals[19] as York finished 11th in the table.[22]
Burn signed for his hometown club, Darlington of the National League North, on 22 June 2018.[24] In his first season he played 17 times, all in league competition, but left the club by mutual consent in October 2019 having failed to add to his appearance tally.[25][26]
Burn signed for Northern League club Whickham on 15 October 2019.[27] He made 12 appearances in all competitions before the season was cut short because of COVID-19.[28]