Buur started playing football at the age of 4, because his big brother also did. The first club he played for was FC Skanderborg, then he joined AGF as a U14 player.
Club career
AGF
At the age of 16 Buur was promoted to the U19 squad. Shortly after that he began training with the first team squad, due to many injuries in the squad. Buur won the 2014 Martin Jørgensens Talent Prize, given by former footballer Martin Jørgensen. Jørgensen said Buur was a player who didn't answer the older players back but instead just respected them. On 25 February 2015, Buur signed a new contract with AGF, and was promoted to the first team squad after playing several friendly matches for them. On 15 March 2015 at the age of only 16, Buur made his official debut for AGF in a Danish 1st Division game against AB and played the whole match. He became the 6th youngest player to debut in the Danish Superliga. After only 10 league games for the club, he didn't extend his contract and left in the summer of 2017.
After leaving AGF, Buur signed for Brabrand IF in the Danish 1st Division on 22 July 2017.
Wolverhampton Wanderers
On 24 August 2017, he signed for then-English Championship club Wolverhampton Wanderers on a two-year deal after an extended trial at the club. He made his first team debut at home to Bristol Rovers on 19 September in the EFL Cup. Buur played the first 90 minutes, and was substituted in the break between normal time and extra time. The game was 0–0 after 90 minutes. Wolves went on to win 1–0 after extra-time. He made his league debut for Wolves, and scored his first professional goal, after coming on as a substitute in a 2–2 draw with Hull City on 3 April 2018, en route to the team winning promotion to the Premier League. On 12 December 2019, Buur made his debut appearance in UEFA European club competition, appearing in Wolves' final 2019-20 UEFA Europa League group stage game against Beşiktaş at Molineux. He provided the assist for Diogo Jota's hat-trick goal. He is signed to Wolves under summer 2023.
