Jenner was born at St Columb Major on 8 August 1848. He was the son of Henry Lascelles Jenner, who was one of two curates to the Rector of St. Columb Major, and later consecrated though not enthroned as the first Bishop of Dunedin and the grandson of Herbert Jenner-Fust. In 1869 Jenner became a clerk in the Probate Division of the High Court and two years later was nominated by the Primate at Canterbury for a post in the Department of Ancient Manuscripts in the British Museum, his father then being the Rector of Wingham, a small village near Canterbury. In 1904, He successfully campaigned for Cornwall to join the Celtic Congress. He jointly founded the Old Cornwall Society at St Ives in 1920 and in 1928 he was a joint founder of the Cornish Gorsedh 
When Jenner was a small boy at St. Columb, his birthplace, he heard at the table some talk between his father and a guest that made him prick up his ears, and no doubt brought sparkles to his eyes which anyone who told him something will remember. They were speaking of a Cornish language. At the first pause in their talk he put his query... 'But is there really a Cornish Language?' and on being assured that at least there had been one, he said 'Then I'm Cornish--that's mine!'
In 1874 Henry Jenner continued his interest in Celtic languages, and in 1875 he read a paper to the Philological Society in London, his subject being the Manx language. The following year he read another paper on the subject of the Cornish language at Mount's Bay. In 1877 he discovered, whilst working in the British Museum, forty two lines of a medieval play written in Cornish around the year 1450.
In 1903 he was made Bard of the Breton Gorsedd, and along with L.C.R. Duncombe-Jewell he jointly founded the first Cornish language society, "Cowethas Kelto-Kernuak". The following year Jenner and Duncombe-Jewell took Cornwall's application for membership of the Celtic Congress, then meeting in Caernarfon. His Bardic name was Gwas Myghal ('Servant of Michael').
In 1904 he published A Handbook of the Cornish language and the Cornish language revival began. His version of Cornish was based upon the form of the language used in West Cornwall in the 18th century, although his pupil Robert Morton Nance would later steer the language revival towards mediaeval Cornish.
At a time when many people thought the Cornish language had died Jenner observed
There has never been a time when there has been no person in Cornwall without a knowledge of the Cornish language ... The reason why a Cornishman should learn Cornish, the outward and audible sign of his separate nationality, is sentimental, and not in the least practical, and if everything sentimental were banished from it, the world would not be as pleasant a place as it is.
In 2010, Michael Everson published a new edition entitled Henry Jenner's Handbook of the Cornish Language, which contains modern IPA phonetic transcriptions to make clear to modern readers what phonology Jenner was recommending. The book also contains three essays written by Jenner thirty years prior to the 1904 publication, as well as some examples of a number of Christmas and New Years cards sent out by Jenner containing original verse by him in Cornish and English.
Jenner married Kitty Lee Rawlings in 1877 (she was a novelist and author of non-fiction under the name Kitty Lee). A biography of Henry and Katherine, including much information about the context in which their work appeared, was published in 2004 by Derek R. Williams.
Jenner was a Tory and Jacobite. He and his wife supported the Order of the White Rose, a society of Stuart sympathizers which he had founded in 1891, and of which he was chancellor. He also actively supported The Royalist, a journal which ran from 1890 to 1905. He was a key figure in the Neo-Jacobite Revival of the 1890s.
After working at the British Museum for more than forty years, in 1909 Jenner and his wife Kitty retired to Hayle, his wife's home town, and in January 1912 he was elected as the Librarian of the Morrab Library, a post he held until 1927. He also served as President of both the Royal Cornwall Polytechnic Society and of the Royal Institution of Cornwall. Jenner had married Kitty Lee Rawlings, herself a novelist and writer on art, in 1877. He died on 8 May 1934 and is buried in St. Uny's Church, Lelant.
Before he died, he said: "The whole object of my life has been to inculcate into Cornish people a sense of their Cornishness."