Furniss was the second son born to William H. Furniss and Mary Elizabeth J. Williams, in Jackson, Mississippi, on January 30, 1874. While still in his youth, his family moved to Indianapolis. His father became the superintendent of the Special Delivery Department of the Indianapolis Post Office. Furniss received his early education in the local city schools and then enrolled in Lincoln University (formerly the Lincoln Institute).
Before his graduation from Lincoln Institute in 1891, Furniss enrolled in the Medical College of Indiana and received his medical degree in 1894, ranking second in a class of fifty-two. Furniss was the only African American in his class. While in medical school, he worked as a clerk for Dr. E. S. Elder, a prominent Indianapolis physician, to pay for his education. On October 26, 1905, he married Lillian Morris.
After competing in a rigorous exam given to fourteen selected candidates from across the country, Furniss served as an intern at the Indianapolis City Hospital and became the first African American professional to serve at the hospital. He then opened a general practice that continued for the next fifty years.
In 1909 Furniss founded the first African American hospital in Indianapolis, Lincoln Hospital, where he served as chief surgeon until its closing in 1915.
Rare for black physicians at the time, Furniss held memberships in American Medical Association, the Indiana State Medical Society, and the Indianapolis Medical Society. He was the state vice-president of the predominantly black National Medical Association. Furniss was also a lifetime member of the Flanner House, an Indianapolis settlement house, and was a charter member of the Senate Avenue YMCA, serving as its first president. In 1917, Furniss was elected to a four-year term on the Indianapolis City Council (1917-1921). Sumner Alexander Furniss died on January 18, 1953.