He attended public schools in London and Minnesota.
Burnham worked as a clerk 1884–1886, then moved to Jackson, Minnesota, in 1887 where he entered the retail shoe business. In 1901 he moved to Spokane, Washington, and worked in real estate and ranching. 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.
Burnham married Neva May Ashley on October 1, 1890, and they had six children, Harold, Percy, Helen, Laurence, Virginia, and Ben. Burnham was elected to the 73rd Congress in 1932, 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 interred in the Greenwood Cathedral Mausoleum, in Greenwood Memorial Park.
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.
Married Neva May Ashley, October 1, 1890 (died 1927). Married second, Florence Kennett, December 25, 1932. Children: Beth Marie (deceased), Harold Ashley, Percy Edmund, Helen Estella (wife of Jefferson Doctorate. Beard, United States Navy), Laurence Malin, Virginia Jean (wife of Robt.