Educated at Eastman"s Royal Naval Academy and Bedford School, Burnett joined the Royal Navy in 1902. He served on the China Station from 1904 and then with the Atlantic and Mediterranean Fleets from 1908. He became an instructor at the Navy Physical Training Schools in 1911.
He served in the First World War and saw action at the Battle of Heligoland Bight in 1914 and at the Battle of Dogger Bank in 1915 and served in destroyers in the Grand Fleet.
He was promoted to lieutenant-commander in April 1918, commander in December 1923 and captain in December 1930. In 1933 he was made Director of Physical Training and Sports.
He was promoted rear-admiral in January 1941 and appointed flag officer of minelayers, and from March 1942 flag officer of the Destroyer Flotillas of the Home Fleet. From January 1943 he was flag officer of the 10th Cruiser Squadron and was promoted to vice-admiral in that role on 9 December 1943.
Flying his flag in HMS Belfast, he saw action in the North Sea and in the Arctic Ocean off the coast of Norway in convoy escort duty, particularly on 26 December at the Battle of North Cape, where he played a major role in the sinking of the Scharnhorst (in most accounts of the battle he is described as a Rear Admiral).
He was Commander-in-Chief, South Atlantic Station from 1944. He became Commander-in-Chief, Plymouth in 1947 and retired from active service in May 1950. He then served as the first director of the White Fish Authority, from its foundation in 1951 until his retirement in November 1954.
He was the younger brother of Air Chief Marshal Sir Charles Burnett.