Born in Prince Edward County, Virginia, Flournoy was educated at Hampden-Sydney College. He engaged as a private teacher and subsequently studied law. He was admitted to the bar and commenced practice in Halifax, Virginia, in 1834.
Flournoy was elected as a Whig to the Thirtieth Congress (March 4, 1847 – March 3, 1849). He was an unsuccessful candidate for reelection in 1848 to the Thirty-first Congress and for election in 1850 to the Thirty-second Congress. He was an unsuccessful candidate of the American Party for Governor of Virginia in 1855, losing to Henry A. Wise.
He served as member of the secession convention in 1861 at Richmond. He then entered the Confederate States Army, raised a company of cavalry, and initially served as its captain. He was promoted to colonel of the 6th Virginia Cavalry. He participated in Stonewall Jackson's 1862 Valley Campaign and saw action at the battles of Port Republic and Cross Keys. He was again an unsuccessful candidate for governor in 1863.
After the war, Fournoy settled in Danville, Virginia, and again practiced law. He served as delegate to the 1876 Democratic National Convention.
He died at his home in Halifax County, Virginia, March 12, 1883, and was interred in the family plot on his estate.