McBurnie played youth football for Bradford City, but played for Manchester United in the 2013 Milk Cup.
McBurnie was given a three-year professional contract by the club on 23 August 2013. In November 2013, Bradford City manager Phil Parkinson stated that he was considering involving McBurnie with the club's first team squad. He made his professional debut on 9 November 2013, in the FA Cup, appearing as a substitute. He later spoke about his desire for further first-team action, and made his first senior start in December 2013.
He was offered a new contract by the club at the end of the 2013–14 season. McBurnie later spoke about his first season as a professional, describing it as "ridiculous." He also spoke about his excitement ahead of the 2014–15 season. In September 2014 manager Phil Parkinson told McBurnie to "step up to the plate" and make the most of the first-team appearances he was making following an injury to regular started James Hanson. In October 2014 Parkinson hinted that McBurnie would be sent out on loan.
McBurnie joined Chester on a one-month loan deal on 22 January 2015. He was recalled early, returning to Bradford on 20 February following injuries to first-team players. Manager Phil Parkinson thought the loan spell had been beneficial to McBurnie's development. He re-joined Chester on loan on 6 March.
On 13 July 2015, McBurnie joined Swansea City on a three-year deal for an undisclosed fee. Later that month he made his first appearances for Swansea in under-21 matches in Holland.
In November 2015 he joined League Two club Newport County on an initial one-month loan. McBurnie made his debut for Newport on 28 November in a league game against Luton Town, scoring a hat-trick after coming on as a 58th-minute substitute. While the loan had been earlier extended until January 2016, he was recalled by Swansea on 24 December 2015. On 7 March 2016, he joined Bristol Rovers on loan for the remainder of the 2015–16 season.
On 23 August 2016, McBurnie scored twice on his debut for Swansea's first team in the second-round 2016–17 EFL Cup tie against Peterborough United. The match finished 1–3 in Swansea's favour. His performance was praised by ex-Swansea player Warren Feeney. McBurnie won the Premier League 2 'Player of the Month' award for January 2017. In November 2016, McBurnie signed a one-year contract extension at the Liberty Stadium until June 2019.
McBurnie appeared as a substitute in Swansea's first three matches of the 2017–18 Premier League. On 31 August 2017, the last day of the English summer transfer window, a loan to Championship club Barnsley for the rest of the season was agreed. However, the transfer was not completed, as paperwork was not filed in time.
In September 2013, McBurnie was called up for a training camp by the Scotland under-19 national team. McBurnie was called up to the Scotland under-19 squad in November 2014. He made his debut in a 1–1 draw against the Netherlands. in April 2015 he was praised by under-19 manager Ricky Sbragia, and subsequently by club manager Phil Parkinson.As of 5 April 2017