Burke, a student at Boston University School of Law, was a reputed runner in the 400 metres and 440 yards, having won the AAU title (440 yards) in 1895. He had no such reputation for the first event he entered in the inaugural modern Olympic Games in Athens, 1896. With many top sprinters absent, Burke surprisingly won the 100 metres. He was also noted for his "crouch start", which was uncommon at that time but in standard use now. His time in the final was 12.0 seconds. In the preliminary heat, he had an even better time – 11.8 seconds.
At the same Olympics, Burke also won the 400 metres, his top event. His times for that event were 58.4 seconds in the preliminary heats and 54.2 seconds in the final, in both of which Burke finished first.
Later in his career, Burke specialised in the longer distances, winning IC4A titles in the 440 and 880 yards events. In 1897, he was one of the initiators of the annually held Boston Marathon, inspired by the success of the marathon event at the 1896 Olympics.