Burn's greatest tournament results were equal first at London 1887 with Isidor Gunsberg (ahead of Joseph Henry Blackburne and Johannes Zukertort), first at Amsterdam 1889 (ahead of a young Emanuel Lasker), second at Breslau 1889 (behind Siegbert Tarrasch), and first at Cologne 1898 (ahead of Rudolf Charousek, Mikhail Chigorin, Carl Schlechter, David Janowski, and Steinitz). He also played at Hastings 1895, the strongest tournament held up to that point, finishing in joint twelfth place with 9½ points out of 21.