He was born as one of the sons of William Burnet. His brother David G. Burnet was one of the first politicians in the Republic of Texas.
After Burnet had in 1791 at the College of New Jersey made his law degree, he moved in 1796 to the Northwest Territory to Cincinnati to work as a lawyer there. In 1814 he was elected to the Ohio House of Representatives and served there for two years. From 1821 to 1828 he worked as an Associate Judge of the Supreme Court of Ohio.
In 1828 he was elected to the U.S. Senate to fill the seat left vacant by William Henry Harrison. He served a total of three years to 1831, but no longer went to for re-election. He was appointed in 1831 by the governments of Kentucky and Virginia in a commission to settle a dispute of the two states with respect to the adopted by Kentucky statute of limitations. Burnet then devoted himself again as a lawyer and later was president of the Cincinnati College and the Medical College of Ohio.
He died in 1853 in Cincinnati.