Burr started his rugby career playing mini-rugby and playing for Aberdeen Wanderers.[1]
He then moved to Gordonstoun, playing rugby for the school.[1]
After a year in Australia, Burr returned to Scotland and played for Aberdeen Grammar.[2][3]
He has also played for the Caledonia academy side.[2] He has also played for a cross-academy select 7s side.[4] The Scottish Rugby Academy side won the Glasgow City 7s tournament.[5]
He has played for one of the two Glasgow Warriors Under 17 sides; the Blacks.[6]
He trained with Glasgow Warriors in the 2016-17 season. Burr stated: "They took me in for training last November because they were short of players and it went from there. I’m not the flashiest player you will ever see but what my academy manager and coaches have said to me is that I’m quite a quick learner. When I was with Glasgow last season during the November Tests and Six Nations windows, I was picking up new things all the time, and able to apply that to my club performances."[11]
Burr graduated to be a Stage 3 player in the BT Sport Scottish Rugby Academy.[12] Stage 3 players are aligned to a professional club and given regional support.[13]
He made his competitive debut for the Warriors away against the Ospreys in the Pro14 on 26 November 2017. He came on as a replacement for Kiran McDonald in a 7 try demolition of the Welsh side.[16][17] He similarly was a replacement the following week in the match against Cardiff Blues.