What consequences may result for driver transporting drunk who has open case of beer in back seat of car?

You have obviously picked the wrong person to trust with your car. Revoke that trust immediately by taking their name off everything. They clearly have exhibited very poor judgment that could end up costing you your car or more. In many states it is illegal to have an open container of any alcoholic beverage in the car, particularly when the vehicle is in operation. In such case, the driver is the one responsible and the consequences can be dire, from heavy fines, to loss of license to jail time.