Is it obligatory to pay for benefiting from the airport lounge buffet designated for business and first-class travelers, which was accessed for free through the father? And does this incur a sin?

It is not permissible to take another person's property without their consent, even if they are a disbeliever. Therefore, if the mediation for your father was done with the permission of the buffet owners or someone authorized to dispose of it, then there is no harm. However, if the permission came from an employee who did not possess such authority, then you have consumed unlawful gains, and you must repent and return the equivalent of the food or its value, because "on the hand is what it has taken until it returns it." This obligation falls upon you, as the liability rests with the direct perpetrator.