Oz was born Richard Frank Oznowicz in Hereford, England to Frances and Isidore Oznowicz, both of whom were puppeteers. His parents were refugees from the Holocaust who moved to England after fighting the Nazis with the Dutch Brigades. Oz's Dutch/Polish father was Jewish and his Flemish mother was a lapsed Catholic. Oz moved to California, United States with his parents when he was five years old and attended Oakland City College.

Oz is known for his work as a puppeteer (including voices), performing with Jim Henson's Muppets. His characters have included Miss Piggy, Fozzie Bear, and Sam the Eagle on The Muppet Show, and Grover, Cookie Monster and Bert on Sesame Street, among many others. The Muppet character Fozzie Bear is actually not named after Frank Oz, as is widely believed. Fozzie is named after Muppet builder Faz Fazakas, who invented the device which enabled Fozzie to wiggle his ears. In addition to performing a variety of characters, Oz has been one of the primary collaborators responsible for the development of the Muppets over the last 30 years. Oz has performed as a Muppeteer in over 75 movies, video releases, and TV specials, as well as countless other public appearances, episodes of Sesame Street, and other Jim Henson series. His puppetry work spans from 1963 to the present day. He also worked with the puppets on the movie Labyrinth, starring David Bowie. Possibly Oz's most famous character is the diminutive Jedi Master Yoda from George Lucas' Star Wars series. Oz performed the voice and puppet for Yoda in The Empire Strikes Back, Return of the Jedi and The Phantom Menace, and provided the voice of the CGI Yoda in Attack of the Clones and Revenge of the Sith. Oz had a great deal of creative input on the character, and was himself responsible for creating the character's trademark style of reversed grammar. George Lucas was so impressed by Oz's performance as Yoda in The Empire Strikes Back that he spent thousands of dollars on an advertising campaign to try to get him an Oscar nomination for Best Supporting Actor.

As an actor, Oz appeared in 1980 as a corrections officer in The Blues Brothers movie, directed by John Landis. He also appeared in later Landis movies An American Werewolf in London, Spies Like Us, Trading Places and Innocent Blood. In 1998, Oz portrayed a warden in Blues Brothers 2000. And while it wasn't for Landis, in 2001 he had a minor part in the Pixar film Monsters, Inc. as Randall's scare assistant Fungus.

Oz began his behind-the-camera work when he co-directed the fantasy film The Dark Crystal with long-time collaborator Jim Henson. The film featured the most advanced puppets ever created for a movie. Oz further employed those skills in directing 1986's Little Shop Of Horrors. The musical film starred Rick Moranis and Ellen Greene, as well as Steve Martin, Bill Murray, John Candy, Christopher Guest, and a 15-foot-tall talking plant (voiced by Levi Stubbs) which at times required up to 40 puppeteers to operate.

Oz went on to direct Dirty Rotten Scoundrels in 1988, starring Steve Martin and Michael Caine, What About Bob? in 1991, starring Bill Murray and Richard Dreyfuss, and HouseSitter in 1992. Later films include The Indian in the Cupboard (1995), In & Out (1997), Bowfinger (1999), The Score (2001), the 2004 re-make of The Stepford Wives, and Death at a Funeral (2007).

Oz has made cameos in several films directed by John Landis, including The Blues Brothers, An American Werewolf in London, Trading Places, Spies Like Us, Innocent Blood, Blues Brothers 2000 and had his name announced over a loudspeaker in Into the Night. Other cameos have included playing a surgeon in scenes cut from the theatrical release of Superman III, The Muppet Movie, The Great Muppet Caper, The Muppets Take Manhattan and several other Jim Henson-related films that didn't involve just his puppeteering.

During the filming of The Score, Marlon Brando refused to take direction from Oz, causing Robert De Niro to act as an intermediary to relay instructions. Brando is reported to called Oz "Miss Piggy", and to have once said to Oz, "I bet you wish I was a puppet so you could stick your hand up my ass and make me do whatever you want".

Oz was considered to direct Charlie and the Chocolate Factory.