Azaria has also voiced the Simpsons characters of Black police officer Lou and the Mexican-American Bumblebee Man. Harry Shearer has played Dr Hibbert, who is Black.
Friday’s statement did not say whether Apu or the other characters would remain on the series.
Bumbling Homer Simpson, housewife Marge, troublemaker Bart, prodigy Lisa and baby Maggie, have captured the changing face of America for more than 30 years in the longest-running scripted show on US television.
The Simpsons is syndicated in more than 100 countries.
Friday’s announcement comes amidst a widespread reckoning for US pop culture about racism following mass protests this month over the killings of Black Americans by police.
Other white actors, including Mike Henry of animated series Family Guy and Kristen Bell of Central Park, have also said they will no longer voice characters of colour.