Burnham was born 1868 in London, England to James and Maria Ann Burnham. He immigrated in 1881 to the United States with his parents, who settled in Spring Valley, Minnesota. He attended public schools in London and Minnesota. Burnham worked as a clerk 1884-1886, then moved to Jackson, Minnesota in 1887 where went the reail shoe business. In 1901 he moved to Spokane, Washington and worked in real estate and ranching.

Burnham moved to San Diego in 1903 and continued to work in real estate with his brother John, until 1917 when he took up banking. Burnham was active in public affairs. Burnham was one of the organizers of the Panama-California Exposition in 1909, serving as vice president from 1909 to 1916. He was also member of the Honorary Commercial Commission to China in 1910, member of the San Diego Library Commission 1926-1932, a member of the San Diego Scientific Library 1926-1932, and vice president of the California-Pacific International Exposition 1935-1936.

Burnham married Neva May Ashley on October 1, 1890 and they had six children, Harold, Percy, Helen, Laurence, Virginia, and Ben. After she died, he married Florence Kennett Dupee.

Burnham was elected to the 73rd Congress in 1934, and reelected in 1934, serving as Representative for Imperial and San Diego counties. He worked to expand Navy presence in San Diego. In 1936, he drew up bills to add 365,000 acres (1460 km²) of federal land in the Carrizo and Vallecito areas to the newly-created Anza-Borrego Desert State Park. In 1936 he did not run again and retired in San Diego, where he lived until his death in 1939. Burnham is interned in Greenwood Cathedral Mausoleum, Greenwood Memorial Park.