Is it permissible for an institute that teaches Arabic to foreigners of various nationalities and religions, and spreads Islam to them directly or indirectly, to continue its work and teaching?

Teaching the Arabic language and spreading Islam is a legitimate and praiseworthy goal. Every means that leads to guiding people and spreading Islam among them is considered permissible, as long as it does not violate the Sharia. There is no objection to teaching the language; in fact, it is one of the best means to attract students to the religion of Islam.