Is it permissible to take out an interest-bearing loan to cover the costs of renovating my elderly mother's house, which is in danger of collapsing, knowing that she is old and does not have sufficient income?

Borrowing with interest is forbidden usury (riba) and is not permissible except in cases of necessity or need that takes the place of necessity. If the harm to the questioner's mother cannot be averted except by borrowing, then there is no objection. But if it is possible by other means, then borrowing with interest is not permissible.