Burwell was the son of Elizabeth Carter Burwell and her first husband,(1680–1721), and was born on June 3, 1720, probably at Fairfield, his father’s plantation on Carter’s Creek in Abingdon Parish, Gloucester County. His grandfathers were (d. 1710), a wealthy Gloucester County planter, and (d. 1732), one of the wealthiest men in North America, who served as president of the governor’s Council. His elder brothers included (d. 1756), who also served as president of the Council, and , who built Carter’s Grove in James City County.
Very little is known about Burwell’s early life. His father died shortly after Burwell’s birth, and in 1724 his mother married a second time, to George Nicholas, a Williamsburg physician. Burwell inherited large estates in Frederick, Isle of Wight, and Prince William counties and perhaps elsewhere, as well as at least one tract in North Carolina. For most of his adult life he lived in Isle of Wight County, which he represented in the House of Burgesses from 1752 to 1758. Burwell was aof Newport Parish, one of the first trustees of the town of Smithfield in 1752, and in 1763 one of the incorporators of the Dismal Swamp Company. About 1742 he married Sarah Nelson, of Yorktown, half sister of William Nelson (1711–1772) and Thomas Nelson (1716–1782), both of whom became presidents of the Council. William Nelson, in turn, married one of Burwell’s sisters, which made Robert Burwell an uncle of Thomas Nelson (1738–1789), a signer of the Declaration of Independence and future governor of Virginia. Burwell and his wife had one son and one daughter.
In April 1762, following the death of Philip Grymes, a member of the Council and a relative by marriage of Carter Burwell, family or business associates in London quietly and quickly arranged for Robert Burwell’s appointment to the Council. When the news reached Williamsburg at the end of July it provoked shock and disbelief because, as Lieutenant Governorput it, of some rumored defect in Burwell’s “mental Qualifications, and an unwarrantable Impetuosity of Temper.” Burwell duly took the oath and was admitted to the Council on July 30, 1762, but at that same meeting his kinsman Thomas Nelson suggested that the Council adopt an address to the king requesting “that he would be graciously pleas’d to appoint some other more able and discreet Person in the Room of Mr. Burwell.” The other councillors postponed the motion, and after Burwell had served on the Council for a few months Fauquier admitted that the doubts about his fitness to serve had subsided. Burwell sat on the Council throughout the remainder of the colonial period.
Burwell had financial problems. Early in the 1770s he attempted to sell some of his land. He eventually had to bequeath a plantation to his son-in-law because he could not raise the £1,000 cash dowry he had promised his daughter at the time of her marriage to John Page (1743–1808), of Rosewell, who was a member of the Council at the end of the colonial period, its first president under the Constitution of 1776, and still later governor of Virginia. Early in March 1770 Burwell’s wife died, plunging him into depression. Both of his children were by then grown and living away from home, and he had nothing to occupy his time except financial worries. Burwell’s loneliness came to an end on the last day of December 1774, when he married Mary Blair Braxton, widow of George Braxton (d. 1761) and a sister-in-law of, who was another signer of the Declaration of Independence and who later served on the Council of State.
Nothing is definitely known about Burwell’s attitude toward the imperial crises that resulted in independence in 1776. Whether because of some lingering uncertainties about his intelligence or temperament, some unrecorded Loyalist leanings, or premonitions of failing health, nobody seriously considered him for any responsible position when the new government of the commonwealth of Virginia was established in July 1776. Burwell lived quietly with his second wife at her Newington estate in King and Queen County until his death there on January 30, 1777. The place of his burial is not recorded.