He played for his local club West of Scotland, in Milngavie, till he was 18. He played for West in the 2022 National Under-18 Youth Cup final at Murrayfield.[2]
He later played for Glasgow Hawks, from where he made his entry to the Glasgow Warriors academy.[3] He was Hawks 2nd XV player of the year in 2023.[4] In the 2024-25 season, while at Glasgow Warriors he was assigned back to Glasgow Hawks in the Premiership draft.[5]
A Warriors fan from boyhood, Burke joined the Glasgow Warriors academy in 2023.[3][6]
He played for the Future XV side in the Super Series in 2023.[7][8]
He played for Glasgow Warriors in the pre-season friendly against Zebre at the Stadio Sergio Lanfranchi in Italy on 30 August 2024. Franco Smith praised the young side despite their defeat. He praised both Burke and Oguntibeju's performance: "I thought Jare [Oguntibeju] and Ryan [Burke] both stood up impressively in the second-row – they are two young men and young athletes that have a lot of potential, and I thought they acquitted themselves very well around the field."[9]
He played in 6 May 2025's Glasgow Warriors U23 match against Benetton Treviso.[10]
He made his competitive debut for Glasgow Warriors on 18 April 2026. He came on as a replacement in the match against Lions at Ellis Park Stadium in South Africa.[11] He became Glasgow Warrior No. 377.
He played for Scotland U20 in 2024.[12] He won the World Rugby Under 20 Trophy with Scotland that year.[6]
He played for Emerging Scotland in their match against Tonga XV in November 2025, coming on as a substitute for Euan McVie after 30 minutes.[13]