Tennant was born in Selby in Yorkshire. His father was Calvert Tennant (named after his grandmother Phyllis Calvert, a granddaughter of Cecilius Calvert, 2nd Baron Baltimore).
His own name derives from his grandmother Rebecca Smithson, widow of Joshua Hitchling. He attended Beverley Grammar School and there is a plaque over one of the entrances to the present school commemorating his discovery of the two elements, osmium and iridium. 
He began to study medicine at Edinburgh in 1781, but after a few months moved to Cambridge, where he devoted himself to botany and chemistry. 
He graduated M.D. at Cambridge in 1796, and about the same time purchased an estate near Cheddar, where he carried out agricultural experiments. 
He was appointed professor of chemistry at Cambridge in 1813, but lived to deliver only one course of lectures, being killed near Boulogne-sur-Mer by the fall of a bridge over which he was riding.