Son of George Burnett (1714-1780) and Helen Burnett (d. 1750). Married (1782) to Christian Leslie (d. 1842), daughter of Prof. John Leslie of Aberdeen, with whom he had at least 6 children.
Burnett was secretary to Sir Andrew Mitchell in Berlin from the early 1760s. Upon Mitchell's death in 1770, Burnett became British Chargé d’Affaires in Berlin, a position which he held until 1772, when he returned back to Scotland.1
Boswell played billiards with Burnett in Berlin on July 15, 1764. Boswell described him as "a very good, solid, clever young fellow". They met on a number of other occasions during Boswell's stay in that city. On August 28 Boswell described him as "a solid, clear-headed fellow, much better than myself", and following dinner with Burnet on September 7 he wrote in his journal that "[t]he civility of Mr Burnet is very great."
One of Boswell's great ambitions during his stay in Berlin was to meet the King of Prussia, Frederick II the Great. Unknown to Boswell, Burnet did nothing to promote a meeting and was secretly against it. Burnet wrote in a letter to Andrew Mitchell that "He [Boswell] teases me to get him presented to [the King of Prssia] but I find he makes such absurd Distinctions between Englishmen and Scotchmen ... that I am certain something very ridiculous would happen on that Occasion. He has told me he must be presented as a Scotchman" 1
- 1Letter from Burnet to Andrew Mitchell dated September 8, 1764. Quoted in Journal of his German and Swiss Travels, p. 99n3