What is the ruling on a bank not notifying a subscriber about an outstanding balance on his Visa card, and his using the card multiple times without payment; and does the bank bear responsibility for that?

The withdrawn amount must be returned to the bank, whether the bank demands it or not. One's liability is not discharged until it is returned. If paying the amount before a certain date is to avoid paying an additional sum due to delay, then this is an unlawful usurious transaction that must be immediately abandoned.