Ammons was the son of Colorado Governor Elias M. Ammons, and was named for his father's friend, U.S. Senator Henry Moore Teller. He served in the United States Army in France during World War I, and then returned to Colorado to work on a ranch and in a newspaper office. He earned a law degree from the University of Denver's Westminster Law School in 1929.
Ammons was elected to the Colorado Senate in 1930 and served until 1935, when Denver Mayor Benjamin F. Stapleton appointed him as Denver city attorney. He was elected governor in 1936. After one two-year term, he was defeated for reelection in 1938 by Ralph L. Carr. Afterward, he practiced law in Denver until his retirement.
Teller Ammons died on January 16, 1972, and was buried in Fairmount Cemetery in Denver, Colorado.