Is it allowed to donate money to an orpahange run by a religion other than Islam where there is religious indoctrination?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

If by your statement about that orphanage that it involves religious indoctrination, you mean that they will spend that money or part of it in that religious indoctrination, then it is not permissible to give optional charity to them because they will use it to disobey Allah by spreading Kufr and teaching it.

But if they will spend the money in orphans needs, such as food, drink and clothing, and so forth, and the orphans are non-Muslims, then in this case, what has to be looked at is whether those donations are obligatory charity, like Zakat or optional charity.

If those donations that are given to the orphanage are obligatory charity, such as Zakat or expiations and the like, then they are not sufficient (i.e. valid) to be paid to non-Muslims, but if it is optional charity, then there is no harm in giving it to that orphanage because there is no harm in giving optional charity to non-Muslims.

Nonetheless, it is more appropriate for a Muslim to give his charity to the Muslims, as there are many needy Muslims around the world.

Imam an-Nawawi said:

"It is desirable (for a Muslim) to give his charity to righteous and virtuous people, honorable and needy ones. However, in case he gives charity to a dissolute person, or a non-Muslim, such as a Jew, a Christian, or a Maggi, it is permissible and he gets a reward in general." [End of quote]

For more benefit, please refer to Fataawa 90749 and 89497.

Allah knows best.