He was a member of the Territorial councils of Ohio from 1799 to 1802 and served in the Ohio State House from 1814 to 1816. Burnet was considered the "father of the Ohio constitution" and was an associate justice of the Ohio Supreme Court from 1821 until his resignation in 1828 to serve as United States Senator. He was elected to the U.S. Senate to fill the vacancy caused by the resignation of William Henry Harrison. He served in the Senate from December 10, 1828, to March 3, 1831.