Is the aforementioned amount considered a debt that must be repaid, and are the heirs obligated to repay it despite their difficult financial circumstances, if the note proving the debt was written only by the creditor without the deceased's signature or knowledge?

As long as the man has not provided evidence to prove the debt owed by your father, he apparently has no right to it. However, if you trust him and are convinced of his honesty, then you must pay the debt as a religious obligation. The debt is due from the father's estate before it is divided. If the father did not leave an inheritance, the heirs are not obligated to pay the debt from their own money. Nevertheless, it is commendable for them to do so if they are able, to absolve their father's responsibility and out of filial piety towards him.