Furnese was the son of Sir Henry Furnese, 1st Baronet, of Gunnersbury House, and his first wife Anne Brough, daughter of Robert Brough. He was educated at Eton College, and spent some time in Germany as a young man.
On 16 December 1708, shortly after his return from the Continent, Furnese was elected Member of Parliament for Truro in a by-election. He held the seat until the British general election, 1710, when he was elected Whig MP for New Romney. He held the seat over four parliaments until 1727. He inherited the baronetcy and a large estate on the death of his father on 30 November 1712. Politically, he was a supporter of Robert Walpole. In the British general election, 1727, he was elected MP for Kent and held the seat through that parliament until his death aged 45 in 1733.
Furnese was married three times. His first wife was his stepsister Anne Balam, the daughter of his father's second wife Matilda Vernon and her first husband Anthony Balam. They married on 1 October 1708 and had one daughter, Anne, who married the Hon. John St John, son of Henry St John, 1st Viscount St John. Anne, Lady Furnese, died on 29 May 1713 aged 25. His second marriage on 8 July 1714 was to Arabella Watson, daughter of Lewis Watson, 1st Earl of Rockingham. She was the mother of Furnese's only son Henry and his second daughter Catherine, who married her first cousin Lewis Watson, 2nd Earl of Rockingham. Arabella died on 5 September 1727. He married thirdly, on 15 May 1729 at Teddington, Middlesex, Lady Anne Shirley, daughter of Robert Shirley, 1st Earl Ferrers. By his last wife, he left a daughter Selina who married Edward Dering. They had one other daughter who predeceased her father.
Furnese died on 7 March 1733 at Waldershare, Kent, and was succeeded by his son Henry. According to a contemporary, he died "by his own fault, for he had one of those colds hanging on him and he drank so hard that he was not sober for ten days before he was taken ill". He was buried at Waldershare on 14 March. His widow survived him by 46 years, dying in Dover Street, London, on 25 February 1779 aged 70. She was buried in Grosvenor Chapel, Westminster.
His only son, Sir Henry Furnese, 3rd Baronet, was born about 1716 and matriculated at Christ Church, Oxford, aged 16 in November 1732. He died unmarried and without issue aged 19 in Montpellier, France, in March 1735. The estate was shared, as co-heirs, by Sir Robert's three surviving daughters: Anne, Catherine (Countess of Rockingham) and Selina. The title became extinct.
- Cokayne, George Edward (1906) Complete Baronetage. Volume V. Exeter: W. Pollard & Co. LCCN 06-23564. pp. 1–2.
- "Parishes: Waldershare", The History and Topographical Survey of the County of Kent: Volume 10 (1800), pp. 50–61. Retrieved 17 November 2010.
- Handley, Stuart (2004) "Furnese, Sir Henry, first baronet (1658–1712)", Oxford Dictionary of National Biography, Oxford University Press, doi:10.1093/ref:odnb/63008 (subscription or UK public library membership required)
- Hastings MSS, vol. III, p. 15, quoted in Handley, Stuart (2002) "FURNESE, Robert (1687-1733), of Waldershare, Kent, and Dover Street, Westminster", in The History of Parliament: the House of Commons 1690-1715, Boydell & Brewer. ISBN 9780521772211
- Hasted, Edward (1798). "Parishes". The History and Topographical Survey of the County of Kent. Institute of Historical Research. 6: 499–531. Retrieved 16 March 2014.
- Newman, A. N. (1970) "FURNESE, Sir Robert, 2nd Bt. (1687-1733), of Waldershare, Kent" in The History of Parliament: the House of Commons 1715-1754, edited by Romney Sedgwick
|Parliament of Great Britain|
|Member of Parliament for Truro
With: Henry Vincent
|Member of Parliament for New Romney
With: Walter Whitfield 1710–1713
Edward Watson 1713–1722
David Papillon 1722–1727
Sir Edward Knatchbull, Bt
Sir Thomas Twisden, Bt
|Member of Parliament for Kent
With: Sir Roger Meredith, Bt
Sir Roger Meredith, Bt
Sir Edward Dering, Bt
|Baronetage of Great Britain|