Born in Enniskillen, now in Northern Ireland, Burkitt attended Dublin University, receiving his BA in 1933 and MB in 1935. Having become a fellow of Edinburgh's Royal College of Surgeons in 1938, Burkitt served in the Royal Army Medical Corps during World War II (1941–46). After the war he worked in Uganda as government surgeon (1946–64), being appointed senior consultant surgeon to the Ugandan Ministry of Health in 1961.

In the late 1950s, Burkitt began studying a form of lymphoma that affected children in his part of Africa. Typically these patients had malignant swellings of the facial bones, although tumors could also be found in the ovaries and abdominal lymph nodes. Burkitt demonstrated that all cases were characterized by infiltration of the affected tissues by lymphocytes, and that the various clinical manifestations were all part of the same cancerous condition, now known as Burkitt's lymphoma.

With his colleagues Edward Williams and Clifford Nelson, Burkitt undertook a geographical survey of the incidence of the disease and found it to be correlated with the same temperature and rainfall zones as malaria. This suggested that the occurrence of the disease may be linked with the distribution of certain insect carriers, as with malaria. Also, an association has now been established between the disease and Epstein-Barr virus, which is isolated from many cases. Burkitt's lymphoma survey is regarded as one of the pioneering studies of geographical pathology.

In later life, and drawing on his experiences of the contrasting diets of developed and developing countries, Burkitt did much to promulgate the benefits to health of a high-fiber diet, and argued that certain diseases of affluent societies, such as bowel cancer and appendicitis, are attributable to dietary fiber deficiency. His publications include Burkitt's Lymphoma (1970), Refined Carbohydrate Foods and Disease (1975), and Western Diseases, Their Emergence and Prevention (1981).