Burney was born in London, the son of the composer and music scholar Charles Burney and his wife Esther Sleepe (c. 1725-1762). He was the brother of Charles Burney and the novelist and diarist Fanny Burney, and half-brother to the novelist Sarah Burney, who kept house for him from 1798 to 1803.
Burney's father obtained him a berth as a midshipman on Cook's Resolution, which sailed for the South Seas in June 1772. Back in England in 1774, he acted as interpreter for Omai, the first Tahitian to visit Britain. He witnessed Cook's killing in Hawaii in 1779. He was belatedly promoted, but in June 1782 commissioned captain of the 50-gun Bristol on a 12-ship convoy to Madras. He saw action as part of Sir Edward Hughes' squadron in the final engagement with the French fleet off Cuddalore on 20 June 1783.
At the end of 1784 Burney fell seriously ill and departed for England. This was the end of his active naval career. Repeated petitions for a new command were rebuffed, in part because of his openly republican political views. However, he became a prolific naval author, who enjoyed the friendship of Charles Lamb, Henry Crabb Robinson and other literary figures.
Burney married Sarah Payne (1759–1832) on 6 September 1785, by whom he had three children, Catherine (1786–1793), Martin Charles (1788–1852), later a solicitor, and Sarah (1796-post 1868). However, he was separated from his wife and living with his half-sister from 1798 to 1803. He was elected a member of the Royal Society in 1809.
In July 1821, aged 71, Burney was promoted to rear-admiral on the retired list after a personal intervention by the duke of Clarence (later William IV), admiral of the fleet. He died on 17 November 1821 and was buried at St. Margaret's, Westminster.
- Chronological History of the Voyages and Discoveries in the South Sea or Pacific Ocean (London, 1803–1817), including History of the Buccaneers of America (1816)
- Chronological History of North-Eastern Voyages of Discovery and of the Early Eastern Navigations of the Russians (London: Payne & Foss, 1819)
- An Essay by Way of Lecture on the Game of Whist (London: privately printed, 1821)
- ODNB entry: Retrieved 28 August 2011. Subscription required.
- Joyce Hemlow etc., ed.: The Journals and Letters of Fanny Burney (London: OUP, 1972), Volume I, p. lxx.
- The Royal Society. This also confirms the biographical information here, as does the epitome in the Concise Dictionary of National Biography, 1939 . For more information on his troubled private life, see The Letters of Sarah Harriet Burney, ed. Lorna J. Clark (Athens, GA/London: University of Georgia Press, 1997), passim; The Journals and Letters of Fanny Burney, esp. Volume 4 (1973). Other information: ODNB entry.
- Quoted in ODNB entry.
- Bibliographical information taken from the British Library Integrated Catalogue: Retrieved 28 August 2011.