He was born at Market Lavington in Wiltshire, and was educated at Queen's College, Oxford, taking holy orders in 1694. The following year, he became chaplain and then fellow of All Souls', and a few years later private chaplain to John Moore, Bishop of Norwich and afterwards of Ely, who appointed him chancellor of the diocese of Norwich. In 1706 he became rector of Thorpe, near Norwich, in 1713 a canon of Ely, in 1724 a canon of Christ Church, Oxford, and on January 23, 1732 Bishop of St Asaph. He died in Oxford, where he had passed most of his life.
Tanner's chief work is his Notitia Monastica, a short account of all the religious houses in England and Wales. This was published at Oxford in 1695; it was reprinted with additions by the author's brother, John Tanner, in 1744; and was reprinted again with further additions by James Nasmith in 1787. He also wrote Bibliotheca Britannico-Hibernica, a dictionary of all the authors who flourished in England, Scotland and Ireland before the opening of the 17th century, at which he laboured for forty years. This was not published until 1748, thirteen years after the author's death. The bishop collected materials for a history of Wiltshire and worked for some time on a new edition of the works of John Leland. His valuable collection of books and manuscripts is in the Bodleian Library at Oxford.