The last time that I checked, while fuel costs does cost a lot for an airline, the marginal cost of the extra fuel to carry a heavy person is quite small...so if they are going to charge, the difference shouldn’t be much, unless they’re scamming their customers (as has been shown for other surcharges).
I suspect there is a reason this isn’t already done. Planes are MASSIVE. Did a quick calculation based on numbers at Wikipedia. If you took a fully loaded 737-300 Southwest airlines plane (so, no first class section to reduce the number of passengers) and filled every single single seat with the average adult American male, the mass of the passengers would be about 25% the total mass of everything. All machines have some start-up inefficiencies — doubling the weight isn’t going to double the fuel consumption because some of that fuel is going to overcoming inefficiencies that are already overcome (friction in the engines, wind resistance — all things that cause fuel consumption but aren’t related to weight), etc. Fuel is not the only cost of operating a plane.
Long story short, the fact that airlines AREN’T doing this already suggests that the increased cost of transporting an overweight flier is less than the cost of implementing such a policy.