That would only be fair if the majority of the fuel burned by the aircraft wasn't burned to overcome the aircraft's air resistance and weight.
As an example, in rough numbers a 737 weighs about 75,000 lbs without passengers or fuel and carries about 130 passengers. That is 576 lbs of aircraft per passenger. Add in maintenance and other operating costs that would still be there is they were flying with no passengers and it becomes obvious that passenger weight is a relatively small percentage of their operating cost.
And, there are also costs due to carrying passengers that have nothing to do with weight. Even if a planeload of passengers weighed 0 lbs, it would still take labor hours, passenger terminal expenses etc to process the weightless passengers onto an off the aircraft.