Heinrich Schomburgk was a tennis player and footballer, winning a German club championship in football in 1906 with VfB Leipzig. In tennis he was German singles champion in 1913, won doubles titles in 1908, 1910, and 1921, and won mixed doubles titles in 1912-13 and 1921. Schomburgk competed at Wimbledon in 1903, losing in the third round of singles and the second round of doubles. He participated in the 1908 London and the 1912 Stockholm Games. In London he was eliminated early in the singles and reached the eighth finals in the doubles with Otto Froitzheim. In Stockholm he earned the gold medal in the mixed together with Dora Köring. In the singles and the doubles with Otto von Müller he again reached the eighth finals.
Schomburgk’s brother Wilhelm, a lawyer, became President of the Deutscher Tennis Bund in the 1930s. Wilhelm was the founder of the club championship matches for senior players of the German Tennis Federation, the Große Schomburgk-Spiele, which are still played today.