Born at Ahmednagar in India, Dewar was educated at Oxford University where he obtained his DPhil in 1942. After research at Oxford he worked in industry as a physical chemist for the Courtauld company from 1945 until his appointment in 1951 as professor of chemistry at Queen Mary College, London. In 1959 Dewar moved to America and served successively as professor of chemistry at the University of Chicago and from 1963 at the University of Texas. In 1990 he was appointed graduate research professor at the University of Florida.

Dewar is noted for his contributions to theoretical chemistry. In his Electronic Theory of Organic Chemistry (1949) he argued strongly for the molecular-orbital theory introduced by Robert Mulliken. He did much to improve molecular-orbital calculations and by the 1960s he was able to claim that he and his colleagues could rapidly and accurately calculate a number of chemical and physical properties of molecules.