Burnet was born at Edinburgh, Scotland in 1643, the son of Robert Burnet, Lord Crimond, a Royalist and Episcopalian lawyer, who became a judge, and of Rachel Johnston, the sister of Johnston of Warristoun, a leader of the Covenanters.
He began his ministry in the rural church at East Saltoun, East Lothian, and served this community devoutly for four years.