Is it permissible to take out a bank loan for another person who is threatened with harm if they do not repay their debt, and would I incur sin for that?

If the problems and harms that would befall your friend, his wife, and his relatives due to his inability to secure a loan reach the level of necessity, and he finds no alternative to alleviate his dire need, and he himself cannot obtain the loan, and you cannot help him except by borrowing with usury, then there is no blame upon you, and we hope you will be rewarded for your intention. A Muslim has a right over another Muslim in دفع الضرورة (averting dire necessity). Thus, jurists have stipulated the obligation to abandon a fard (religious obligation) or commit a haram (forbidden act) to save another Muslim, such as a fasting person breaking their fast to save a drowning person, or swearing falsely to save a Muslim from perdition.