Moore was born in Los Angeles, California, the son of Nora Eileen (Orr) and John Richard Moore, Sr., a banker. His mother was Irish, and his paternal grandparents were from England and Ireland. He made his film debut in 1927 in the silent film The Beloved Rogue, where he portrayed silent film star John Barrymore's character as a one-year-old baby. At the time of his death, Moore was one of the last surviving actors to have appeared in silent film. He quickly gained notable supporting roles. He had a significant role as Marlene Dietrich's son in Josef von Sternberg's drama Blonde Venus (1932). He also appeared with Barbara Stanwyck in So Big (1932), with Walter Huston in Gabriel Over the White House (1933) and with Spencer Tracy in Man's Castle (1933).

Besides appearing in a number of major feature films, he was featured as a regular in the Our Gang series during the 1932–1933 season. Although he only played in eight Our Gang films, in those films he played an important role as the leader of the gang. He left the series after one year to play in more feature films. In addition to his Our Gang work, Moore is most remembered for his portrayal of the title character in the 1933 adaptation of Charles Dickens' Oliver Twist. In 1935, he played the historical role of Joseph Meister in the film drama The Story of Louis Pasteur about the life of scientist Louis Pasteur. In 1941, he portrayed the brother of Gary Cooper in the war drama Sergeant York under the direction of Howard Hawks. He is also famous for giving Shirley Temple her first romantic onscreen kiss, in the film Miss Annie Rooney.

Moore was less successful as a teenage actor and young adult and he often had to play in B-movies such as Dangerous Years during the 1940s. One of his last notable film roles was in Out of the Past (1947), in which he portrayed Robert Mitchum's deaf young assistant, "The Kid". Moore played his last role as a young soldier in Eight Iron Men (1952). He later performed on Broadway, in stock and on television. He went on to teach and write books about acting, edit Equity News, and produce an Oscar-nominated short film (The Boy and the Eagle), and industrial films. He retired from acting in the late 1950s. In 1966, after battling alcohol and drugs, he founded a public relations firm, Dick Moore and Associates, which he ran until 2010.

In 1984, Moore published Twinkle, Twinkle, Little Star (But Don't Have Sex or Take the Car), a book about his and others' experiences as child actors. Moore owned a public relations firm, Dick Moore and Associates. Founded in 1966, it existed for 44 years. From 1988 until his death in 2015 Moore was married to the Actress Jane Powell. The two met when Moore interviewed Powell for Twinkle, Twinkle, Little Star. The couple lived in Manhattan and Wilton, Connecticut.

In March 2013, Powell reported that Moore had arthritis and "bouts of dementia".

Moore died at a hospice near Wilton, Connecticut on September 7, 2015, five days before his 90th birthday.