He was born in New York City, son of Patrick Naish, who had emigrated from County Limerick, Ireland in about 1890. Patrick was a nephew of John Naish, Lord Chancellor of Ireland. Naish appeared on stage for several years before he began his film career. He began as a member of Gus Edwards's vaudeville troupe of child performers. After World War I, Naish formed his own song and dance act in Paris. He was traveling the globe from Europe to Egypt to Asia, when his China-bound ship developed engine problems, leaving him in California in 1926.

Naish's uncredited bit role in What Price Glory? (1926) launched his career in more than two hundred films. He was twice nominated for an Academy Award for Best Supporting Actor, the first for his role as Giuseppe in the movie Sahara (1943) in which he delivers one of the most moving speeches in any wartime film:

Naish was married (from 1929 until his death) to Actress Gladys Heaney (1907–1987); they had one daughter, Elaine.

He often played villains from Gangsters in numerous Paramount pictures to mad Scientists, such as Dr. Daka in the Batman film serial. In the 1940s Naish was a supporting character in a number of horror films. He played Boris Karloff's assistant in House of Frankenstein (1944).

The second was for his performance as the title character's Hispanic father in the movie A Medal for Benny (1945). For the latter film, he won the Golden Globe Award for Best Supporting Actor – Motion Picture.

On radio, Naish starred as Luigi Basco on the popular CBS program Life with Luigi (1948–1953). Luigi's popularity resulted in a CBS television series of the same name, with Naish reprising his role.

In 1955, Naish originated the role of Alfieri in the one-act, verse version of Arthur Miller's A View from the Bridge on Broadway, also starring Van Heflin and Eileen Heckart.

In 1971, he appeared in his final film role, Dracula vs. Frankenstein (1971), as a mad scientist; a role descended from the original Dr. Frankenstein takes to murdering young women for experimentation in hopes of reviving his ancestor's creation, with help from his mute assistant, played by Lon Chaney Jr., whose film appearance was also his last.

Naish retired to San Diego, CA and died of emphysema on January 24, 1973, at Scripps Memorial Hospital in the northern community of La Jolla, California, three days after his 77th birthday. He is interred at Calvary Cemetery in East Los Angeles, California. For his contributions to television, he has a star on the Hollywood Walk of Fame at 6145 Hollywood Boulevard.

Of Irish descent, he rarely played an Irishman, explaining, "When the part of an Irishman comes along, nobody ever thinks of me." He portrayed numerous other ethnicities including Southern European, Eastern European, Latin American, Native American, Middle Eastern, South Asian, East Asian, Southeast Asian, Pacific Islander—even African American, which earned him the moniker "Hollywood's one-man U.N.".