Must the heirs pay a sum to a person who paid it to the tax office on their behalf, knowing that this person was not authorized to do so, and that the payment was to enable him to make changes to the shop which was owned by the heirs?

If a person pays a due right on behalf of another, intending to claim it back from them, then they have the right to demand its repayment, whether the person on whose behalf it was paid gave permission or not. This is the view of the Hanbalis and Malikis. It should be considered whether what he paid was a right that was binding upon you and he intended to claim it back, in which case he has the right to demand it. Or if it was not binding upon you and he did it for his own benefit, then he does not have the right to claim it back. Issues of disputes require direct consultation with scholars.