Burns is often described as the father of the modern Democratic Party of Hawai'i. From 1948 he held various leadership positions in the territorial Democratic Party, culminating in being chair of the territorial party in 1952. He is credited with building a coalition with organized labor and Japanese-Americans to strengthen the Democratic Party.
In 1956 he was elected Delegate from Hawaii. As Delegate he played a key role in lobbying for Hawaii statehood, a goal that was achieved on March 12, 1959, when the statehood bill was signed by President Dwight D. Eisenhower. He sought to become the first Governor of the newly formed State of Hawaii, but lost the election to then-Territorial Governor William F. Quinn.
Three years later in 1962, Burns won the election to become Governor. He was re-elected in 1966 and 1970, each time with a different Lieutenant Governor as his running mate. Burns became ill to the point of incapacity in October 1973 and Burns' third elected Lieutenant Governor, George R. Ariyoshi, became Acting Governor through the end of Burns' third term, before Ariyoshi was elected in his own right as Governor for the term beginning in 1974.
The John A. Burns School of Medicine, an institution of the University of Hawaii at Manoa, is named for him. In a controversial move in 1997, Governor Benjamin J. Cayetano, named the newly completed Interstate H-3 in Burns' honor.