Tinker was born in Stamford, Connecticut, the son of Margaret (née Hessin) and Arthur Almerin Tinker.
During World War II, Tinker served in the United States Army Air Forces Reserve. He graduated from Dartmouth College. His sons, Mark and John, are also producers. In 1961, Tinker rejoined NBC and was the head of West Coast programming, where he developed I Spy, Dr. Kildare, and The Man from U.N.C.L.E.
Tinker married Moore in 1962. In 1969, they formed the television production company MTM Enterprises. Tinker hired Room 222 writers James L. Brooks and Allan Burns to create and produce the company's first television series, The Mary Tyler Moore Show. MTM produced such popular American sitcoms and drama television series as Rhoda, The Bob Newhart Show, WKRP in Cincinnati, Hill Street Blues, and St. Elsewhere. After his divorce from Moore in 1981, Tinker left MTM to become the chairman and CEO of NBC, then the perennial last-place American television network (in terms of Nielsen ratings and profits). During Tinker's tenure in NBC's top position, the network regained ratings and produced the popular shows The Cosby Show, Family Ties, The Golden Girls, Cheers, Night Court, and Hill Street Blues. Tinker left the network in 1986, shortly after its parent company RCA was bought by General Electric. After leaving NBC, Tinker tried to repeat his success with MTM by forming GTG (Grant Tinker-Gannett) Entertainment, but the business venture failed and the company closed in 1990.