Mr Gergiev is the music director of the Mariinsky Theatre in St Petersburg, Russia, and its White Nights Festival, as well as chief conductor of the Munich Philharmonic.
He has been dropped by his management company over his ties to Russian President Vladimir Putin.
In a statement, the EIF said: “The board of trustees of the Edinburgh International Festival has asked for, and accepted, the resignation of Valery Gergiev as honorary president of the festival.
“Edinburgh is twinned with the city of Kyiv and this action is being taken in sympathy with, and support of, its citizens.”
EIF said the resignation is effective immediately.
Munich mayor Dieter Reiter has set a Monday deadline calling on Mr Gergiev to publicly denounce the Russian invasion.
If Mr Gergiev does not comply, Mr Reiter has said he will remove him as chief conductor of the Munich Philharmonic.
