Burgess was educated at Winchester, and at Corpus Christi College, Oxford. In 1783 he became a fellow of this college, and in 1785 was appointed chaplain to Shute Barrington, Bishop of Salisbury. In 1803 he was appointed to the vacant bishopric of St David's, which he held for twenty years. He founded the Society for Promoting Christian Knowledge in the diocese, and also St David's College at Lampeter. In 1820 he was appointed first president of the recently founded Royal Society of Literature. In 1823 he was promoted to the see of Salisbury, over which he presided for twelve years.
John Scandrett Harford, The Life of T. Burgess, Bishop of Salisbury, London, 1840; Dictionary of National Biography, Oxford, 1997.