Is it permissible to sell an apartment in a non-Islamic country to a non-Muslim buyer who will take out an interest-based loan from a bank, for the purpose of emigrating to an Islamic country? And is this situation considered a necessity that permits it?

You may sell your house to a prospective buyer even if he will pay the price with an interest-bearing loan, because after the loan is taken, it becomes a debt fixed in his responsibility. What is forbidden are the usurious interests resulting from it, not the principal of the loan itself, due to the Almighty's saying: {Baqarah:279}. It is permissible to sell to a Muslim or a non-Muslim, unless it is proven or highly probable that the buyer will use the house for something forbidden.