Robert Burn was born on 22 October 1829, the second son of Andrew Burn (1790/91–1874), rector of Kynnersley, Shropshire, and his second wife, Mary Harris (c 1792-1843). He attended Shrewsbury School under Benjamin Hall Kennedy and was admitted to Trinity in 1848, graduating Senior Classic in 1852. He was elected a Fellow of Trinity in 1854, and for many years he lectured on classical subjects; from 1856 to 1872 he was a tutor and Dean from 1861 to 1863. He vacated his fellowship on his marriage in 1873 to Augusta Sophia Prescott (1835–1915). Re-elected a fellow of Trinity in 1874, he was also praelector in Roman archaeology from 1873 to 1885. He was ordained deacon in 1860 and priest in 1862, and received an honorary LL.D from Glasgow University in 1883.
Burn was one of the first Englishmen to study the archaeology of the Rome and the Campagna, which he frequently visited during vacations. His publications included Rome and the Campagna (1871), Old Rome (1880), Roman Literature in Relation to Roman Art (1888), and Ancient Rome and its Neighbourhood (1895). He was an original member of the Governing Body of Winchester College in 1871. In 1881 he was president of the Cambridge Antiquarian Society.
Burn was a member of the Alpine Club from 1860 to 1867, one of the first captains of the Cambridge University rifle corps, and among the committee of Trinity men who drew up the Cambridge University rules for football in 1863. During the last twenty years of his life, however, he was an invalid confined to a bath chair. He died on 30 April 1904 at his home and was buried in St Giles's cemetery at Cambridge. There is a brass to his memory in the ante-chapel of Trinity College.