Shaw was born at Bierton, Buckinghamshire, and was educated at Magdalen Hall, Oxford, receiving his M.A. in 1772. He took up the profession of medical practitioner. In 1786 he became the assistant lecturer in botany at Oxford University. He was a co-founder of the Linnean Society in 1788, and became a fellow of the Royal Society in 1789.
In 1791 Shaw became assistant keeper of the natural history department at the British Museum, succeeding Edward Whitaker Gray as keeper in 1806. He found that most of the items donated to the museum by Hans Sloane were in very bad condition. Medical and anatomical material was sent to the museum at the Royal College of Surgeons, but many of the stuffed animals and birds had deteriorated and had to be burnt. He was succeeded after his death by his assistant Charles Konig.
Shaw published one of the first English descriptions with scientific names of several Australian animals in his "Zoology of New Holland" (1794). He was among the first scientists to examine a platypus and published the first scientific description of it in The Naturalist's Miscellany in 1799.
His other publications included: