Henry C. Burnup was the son of Henry Burnup and his wife Margaret, born Pollock. Both his parents died before he was seven years old and he was brought up by a maiden-aunt, Miss A. Smith, who was interested in plants and a shell collector. He was educated in Edinburgh, emigrated to Natal in 1874 and established an accountancy business in Pietermaritzburg. When Miss Smith died in 1885 she left her shell collection to him and conchology became his life-long passion. His main interest was in land snails, of which he donated a small collection to the South African Museum in 1895. Most were collected in Pietermaritzburg, but others on trips to different parts of KwaZulu-Natal, particularly near easily accessible places such as railway stations. However, he also collected marine shells during several visits to the coast each year - even though a stay at the coast often brought on a severe attack of eczema.
On many of his early collecting trips Burnup was accompanied by John F. Queckett*, and they were the first collectors to obtain shells from the stomach of the Musselcracker fish which they obtained from a Durban fishmonger, Alex ("Lexy") Anderson. Burnup also inspired or befriended the collectors Charles W. Alexander*, H.P. Thomasset*, William Falcon*, H.J. Puzey*, and others. His shell collection eventually became the largest ever assembled by one person in South Africa and included many type specimens of new species. Initially he sent most of his marine specimens to the British conchologist J.H. Ponsonby for identification, but after the latter's death in 1916 much of his material went to J.R. le Brockton Tomlin. For the non-marine molluscs he consulted also M. Connolly*, J.C. Melvill, and his own nephew, Hugh Watson*. Burnup was Natal's first local malacologist, responsible for the discovery of more species of land snails and slugs in KwaZulu-Natal than any other person, and was the most important personality to drive the study of its molluscs.
Burnup retired early, around 1903, and for the rest of his life served as honorary keeper of the conchological collections at the Natal Museum in Pietermaritzburg, directed by Dr E. Warren*. He donated his collection to the museum, including much material collected by others who were inspired by his enthusiasm. In 1904 he and Warren undertook a collecting trip to Hlabisa, just north of the Umfolozi River, and other trips followed. He sent a number of pulmonates to H.A. Pilsbry at the Academy of Natural Sciences in Philadelphia, for inclusion in the Manual of conchology - the only attempt so far to monograph the snails of the world. In 1913 he presented shells to the McGregor Museum in Kimberley and, working at the Natal Museum, identified the McGregor Museum's acquisitions of molluscs.
During his retirement he published eight papers dealing with various groups of land snails, including descriptions of new species. Several of his papers appeared in the Annals of the Natal Museum: "On Afrodonta..." (1912, Vol. 2(3), pp. 333-343), "On South African Enneae..." (1914, Vol. 3(1), pp. 29-83), "The genus Sculptaria Pfeifer..." (1923, Vol. 5(1), pp. 1-44), and "On some South African Gulellae..." (1925, Vol. 5(2), pp. 101-158; 1926, Vol. 5(3), pp. 351-398). The genus burnupena of common intertidal Cape gasteropods was named in his honour by Iredale in 1918. He was also commemorated in the genus Burnupia of freshwater limpets, and in many species names.
Burnup was a rather formal and scholarly person. He served on the council of the Natal Society for 1898/9, and on its museum committee from about 1890 to at least 1898. The Society's museum became the Natal Museum in 1903, with Burnup as a member of the Board of Trustees. In 1916/7 he became a foundation member of the South African Biological Society. In 1875 he married Alice M. Humble, who died in 1919.