He was very sharp.”[4] He was born into a secular Jewish family[5] in San Francisco, where in the third grade he was chosen as a subject of Stanford University professor Lewis Terman's study of gifted children.

"[6] During his junior year at Stanford during the 1930s, Oppenheimer visited the studios of radio station KFRC in San Francisco, and soon started spending all his spare time there.

He made his broadcast debut performing a comedy sketch he'd written on the station's popular coast-to-coast comedy-variety radio program, Blue Monday Jamboree.

As a staff writer on those programs, Oppenheimer wrote sketch comedy for many Hollywood stars, including Fanny Brice, George Burns and Gracie Allen, Bing Crosby, Marlene Dietrich, Judy Garland, Bob Hope, and Ginger Rogers.

Stark immediately hired Oppenheimer to write for the popular radio program, The Baby Snooks Show, which starred Brice as a wise-beyond-her-years little girl who constantly drove her daddy crazy.

[4] In 1948, shortly after The Baby Snooks Show went off the air, CBS asked Oppenheimer to write a script for a new unsponsored radio sitcom, My Favorite Husband, starring Lucille Ball.

[4] Oppenheimer decided to make her radio character more like Baby Snooks: less sophisticated, more childlike, scheming, and impulsive—taking Lucy and the show in a new direction, with broad, slapstick comedy.

[4] Oppenheimer left I Love Lucy in 1956 to take an executive post at NBC, where he produced a series of TV specials, including the General Motors 50th Anniversary Show (1957), Ford Startime (1959), The Ten Commandments (1959), and the 1959 Emmy Awards.

[9] Oppenheimer met his future wife, Estelle (née Weiss), in 1942 while she was working as the manager of the Popular Records Department at Wallichs Music City, on the corner of Sunset and Vine in Hollywood.

[10] Oppenheimer was also an inventor, holding 18 patents covering a variety of devices, including the in-the-lens teleprompter, first used on television by Lucille Ball and Desi Arnaz for a filmed Philip Morris cigarette commercial which aired on I Love Lucy on December 14, 1953.

In the 2021 Aaron Sorkin film Being the Ricardos, Oppenheimer is portrayed as head writer and producer of I Love Lucy by Tony Hale, and as an older man by John Rubinstein.