Is the money due to the heirs from the bank after the death of their father—knowing that it is an interest-based bank—lawful or unlawful, given that they were unaware of this entitlement beforehand?

If the deceased left an amount as a deposit in an interest-based bank, and the heirs do not know that its source is unlawful, then it is permissible for them to benefit from it, and it is added to the inheritance. It is lawful by the consensus of scholars, unless there is a sign of its unlawfulness. However, if they know that its source is usurious interest, then it is not permissible for them to benefit from it, and they must dispose of it by spending it on the welfare of Muslims or giving it to the poor and needy.