He was a younger son of Sir Thomas Rawlinson (1647-1708), Lord Mayor of the City of London in 1705-6, and a brother of Thomas Rawlinson (1681-1725), the bibliophile who ruined himself in the South Sea Company, at whose sale in 1734 Richard bought many of the Orientalia. He was educated at St Paul's School, at Eton College, and at St John's College, Oxford. In 1714 he was elected a Fellow of the Royal Society, where he was inducted by Newton. In 1716 he was ordained, but as he was a nonjuror (see Nonjuring schism) and Jacobite, the ceremony was performed by a non-juring bishop, Jeremy Collier.
Rawlinson then travelled in England and on the continent of Europe, where he passed several years, making very diverse collections of manuscripts, coins and curiosities, his books alone forming three libraries, English, foreign and Classical. In 1728 he became a bishop among the nonjurors, but he hardly ever appears to have discharged episcopal functions, preferring to pass his time in collecting books and manuscripts, pictures and curiosities. He died at Islington, London.
Towards the end of his life he quarrelled with both the Royal Society and the Society of Antiquaries. Cutting the Society of Antiquaries from his bequests, he began transferring his collections to the Bodleian. A series of almanacs in 175 volumes, ranging in date from 1607 to 1747 arrived in 1752-55. At his death Rawlinson left to the Library 5,205 manuscripts bound in volumes that include many rare broadsides and other printed ephemera, his curiosities, and some other property that endowed a professorship of Anglo-Saxon at Oxford. He was also a benefactor to St John's College, Oxford.