Is it permissible to take out a loan with interest from an Islamic bank, or government financing with similar terms, to cover family expenses and sustain a project, given the inability to repay current debts?

If a loan involves a stipulated increase, it is impermissible to proceed with it, whether the lender is a government entity or a bank. Borrowing with usury is one of the gravest sins, incurring curses and removing blessings. It is only permissible in cases of extreme necessity, such as fear of perishing.

Paying off a debt that one is unable to settle is not considered a necessity, unless delaying the debt payment inevitably leads to imprisonment for the debtor, and there is no other way to pay it off except through usurious borrowing. In such a case, borrowing only the amount of that specific debt is permissible.

However, purchasing a commodity with deferred payment and then selling it for immediate cash due to need (tawarruq) is permissible.

And whoever places their trust in Allah and fears Him, He will provide them with good sustenance and suffice them in what concerns them.