In 2008, while playing for the Canterbury-Bankstown Harold Matthews Cup team, Burr represented the New South Wales under-16 team.[4] In 2009, he made his debut for the Bulldogs' under-20 side at 16 years old and represented the New South Wales under-18 team.[5][6] In 2010, he represented New South Wales under-18 again, scoring a try in their 22–18 win over Queensland.[7] Burr played for the Bulldogs' under-20 side until 2012, playing 73 games for the team.[8][9]
In 2013, Burr moved up to the Canterbury-Bankstown New South Wales Cup team, winning the club's NSW Cup Player of the Year award.[10] In Round 25 of the 2013 NRL season, Burr made his debut for Canterbury-Bankstown in a 34–14 win over the Penrith Panthers.[11]
In 2014, Burr did not appear in first grade for the Canterbury club. On 21 September 2014, Burr was named on the bench in the 2014 New South Wales Cup Team of the Year.[12]
Gold Coast Titans
On 4 September 2014, Burr signed a two-year contract with the Gold Coast Titans starting in 2015.[13] In Round 1 of the 2015 NRL season, Burr made his debut for the Gold Coast, coming off the bench in a 18–19 loss to the Wests Tigers.[14] In Round 15 of the 2015 NRL season, he scored his first NRL try in a 14–36 loss to the New Zealand Warriors.[15] In his first season at the club, Burr played 20 games, starting eight games at lock.[16]
On 27 October 2016, Burr signed with the Bradford Bulls. In January 2017, following Bradford's liquidation, he signed a one-year deal with the Leigh Centurions.[19][20]
In 2018, Burr returned to Australia, re-signing with the Canterbury-Bankstown club and played for their New South Wales Cup team. On 7 September 2018, he was named the Canterbury-Bankstown NSW Cup Player of the Year.[22] On 23 September 2018, he came off the bench in Canterbury's NSW Cup Grand Final win over Newtown. The following week, he played for Canterbury in their NRL State Championship win over Redcliffe at ANZ Stadium.[23][24]
New Zealand Warriors
On 19 November 2018, Burr signed with the New Zealand Warriors on a two-year contract.[25] In Round 1 of the 2019 NRL season, he made his debut for the Warriors, starting at lock in a 40–6 win over Canterbury.[26] He played 23 games in his first season with the club, starting 13 at lock.