Is the questioner obligated to pay off his father's debts, knowing that these debts may have resulted from gambling?

The fundamental principle is that one's financial liability (dhimmah) is free from debt, and it is only engaged by valid evidence (bayyinah), not by mere claim, doubt, or probability. The proof for this is what 'Izz al-Din ibn 'Abd al-Salam stated in "Qawa'id al-Ahkam" and what al-Juwayni mentioned in "Ghiyath al-Umam." What the questioner mentioned does not constitute evidence by which a debt can be established.