Is it permissible for the husband to pay for his wife's medical treatment in another country at the expense of the state where he resides, knowing that the cost of treatment is very high and she cannot afford it?

The basic principle is that people's wealth is inviolable, and it is not permissible to assault it or take it without the consent of its owners, whether they are individuals or institutions. Therefore, if the entity providing treatment for your wife only allows it for residents, then it is forbidden for you to take it from them; due to the saying of the Prophet, peace and blessings be upon him: "Indeed, your blood, your wealth, and your honor are sacred among you."