What is the ruling on swearing that the deceased paid off the debt, even though the lender has forgiven it, and is this oath considered a sin?

May Allah reward you for forgiving the debt of the deceased. As for false oaths, the general rule is that they are forbidden, unless they are for averting a significant harm, and that harm cannot be averted otherwise. Examples of this include saving an innocent life from perishing, or concealing a trust from an oppressor. However, if it is possible to use ma'areedh (ambiguous speech) and tawriyah (equivocation), that is preferable. Therefore, you must repent and seek forgiveness. There is no expiation for a yameen al-ghamoos (a deliberately false oath sworn to usurp a right or to conceal a'sin) according to the majority of scholars, while Imam Shafi'i holds that expiation is obligatory, and this is more cautious.