Is the money remaining from the sale of a house purchased with an interest-bearing bank loan, after settling the remainder of the loan, considered lawful (halal) or unlawful (haram) money?

You must pay off the remaining loan. If it is possible to avoid paying the usurious interest without harm, then do so. Otherwise, pay off everything quickly and hasten to sincere repentance, for usury is one of the gravest major sins. The process of purchasing the house is valid, and the house is yours. You can sell it for any price you wish, whether immediate or deferred. Any profit, rent, or dwelling you obtain from it is lawful for you.