BRASÍLIA—Brazil’s Supreme Court ordered the country’s house speaker, who has led the impeachment drive against President Dilma Rousseff, to relinquish his post while he faces corruption charges, adding more uncertainty to the nation’s turbulent political landscape.

The order requires Eduardo Cunha to step down immediately from his influential job as speaker of the lower house of congress and relinquish his position as a federal deputy. Mr. Cunha was charged last year with pocketing millions in funds diverted from Brazil’s...