Is it permissible to mortgage someone else's house to obtain a loan from an Islamic bank, and what is the evidence for that?

It is permissible to mortgage a borrowed item. So, if your relative agrees to lend you his house for you to mortgage it, that is permissible. However, it is not permissible for him to take money in return for that, because a loan (of an item for use, ‘ariya) is among the contracts of gratuities. Similarly, it is permissible for your relative to mortgage his house on your behalf as a guarantee or surety, and this is among the contracts of gratuity and benevolence. It is not permissible for him to request a fee in exchange for his surety.