J. Dampier Green was educated at Chester College and commenced his engineering career on the Dee reclamation works on the Dee Estates, Chester, of which he was a part owner. He was also the owner of lead, copper and coal mines and other properties in north Wales. In 1886 he came to the Cape Colony to work on the construction of the Cape Central Railways. When his work was completed he went to the Malmani gold fields in the western Transvaal. While there he became involved in a plan to invade Matabeleland (now part of Zimbabwe), but owing to opposition by the British government the plan was abandoned.
Green was a Fellow of the Geological Society of London. He resided in Johannesburg from 1895 or earlier to at least 1910. In 1895 he served on the first council of the Geological Society of South Africa and remained a member of council for at least the next two years. When the society resumed its activities after the Anglo-Boer War he served as honorary secretary and honorary editor of its Transactions for 1902/3. He was still the honorary secretary of the society in 1910, but was no longer a member in 1915. His main scientific contribution during these years was a paper, "Discussion on glaciers in South Africa", which was published in the society's Transactions (Vol. 4, pp. 171-172) for 1898-1899. It dealt, among others, with ice scratching observed in the Outeniqua Mountains. Later he participated in the discussion of a paper by A.R. Sawyer* on the geology of Chuniespoort, northern Transvaal, published in the Transactions of the Institution of Mining Engineers (Newcastle-upon-Tyne) in 1905-1906.
By 1903 Green had joined the South African Association for the Advancement of Science and was still a member in 1906. He was declared insolvent in 1905. At his death in 1926 he was survived by his wife, Eliza L. Plastow.