Tanner was born in Frome, Somerset, the eldest son of Joseph Tanner. He was educated at Mill Hill School, London, and at St John's College, Cambridge, where he took a First in the Historical Tripos in 1882. He was President of the Cambridge Union Society in Easter Term, 1883. He was a lecturer in History at St John's, from 1883 to 1921, and lecturer on Indian History to Indian Civil Service students, from 1885 to 1893.

In 1883 Tanner became a Fellow of St John's and was an Assistant Tutor from 1895 to 1900, a Tutor from 1900 to 1912, and Tutorial Bursar, 1900–21. He served as a deputy to the Regius Professor of Modern History, 1926–27.^[4]

In 1888 Tanner married Charlotte Maria, second daughter of George J. Larkman of Belton. After living in Cambridge for forty-two years he moved to Woodside, Aldeburgh, Suffolk. He is buried at the Parish of the Ascension Burial Ground in Cambridge.

• Two Discourses of the Navy 1638 and 1659, by John Hollond; and A Discourse of the Navy, 1660, by Sir Robert Slyngesbie, ed., 1896.
• Pepys's Memoirs of the Royal Navy, ed., 1906.
• The Navy of the Commonwealth and the First Dutch War, 1906.
• The Historical Register of the University of Cambridge, being a supplement to the Calendar with a record of university offices, honours and distinctions to the year 1910, ed., 1917.
• Samuel Pepys and the Royal Navy [Lees Knowles Lectures, 1919], 1920.
• Tudor Constitutional Documents, A.D. 1485–1603, ed., 1922.
• Mr Pepys: An Introduction to the Diary together with a Sketch of his Later Life, 1925.
• Pepys's Naval Minutes, 1926.
• Private Correspondence of Samuel Pepys, 1679–1703, ed., 1926.
• English Constitutional Conflicts of the seventeenth century, 1603–1689, 1928.
• Further Correspondence of Samuel Pepys, 1662–1679, ed., 1929.
• Constitutional Documents of the Reign of James I, A.D. 1603-1625, 1930.

• Joseph Robson Tanner at Find a Grave