Burnett was born in Montrose, Scotland on 16 January 1779 and attended Montrose Grammar School. He was appointed surgeon's mate on board the "Edgar" soon after his arrival at Edinburgh to pursue his medical studies. Later he served as assistant-surgeon in the "Goliath" under Sir John Jervis, and was present at St. Vincent and the siege of Cadiz. He also served with distinction at battles of the Nile and Trafalgar.