Burton was the eldest of two sons born to politician Francis Burton of County Clare by his wife, Mary ("née" Conyngham). His paternal grandfather, also named Francis Burton (1640–1714), sat in Irish parliament for Ennis from 1691–1714. Originally a branch of the Musards, Lords of Stavely, the Burton family settled in Richmond, Yorkshire after the Norman conquest. Sir Edward Burton (1442–1524), knighted by Edward IV in 1460 after the Second Battle of St Albans, settled in Longnor, Shropshire. His descendant Thomas Burton moved to Ireland in 1610.