I think that the additional weight on the water on the surface of the outer airplane body increases friction with the air, and also weight of the aircraft. But does the fuel consumption increase? And by how much?
Former C-130 Flight engineer here, it was my job to calculate performance data for takeoff and landing.
Rain only matters because it increases stopping distance. It doesn’t affect engine performance, and it never even occurred to me that a wet airplane would be heavier than a dry one. To get a sense of weights we care about, our empty weight is ~90k pounds and our max (peacetime) takeoff weight is 155k pounds. Performance numbers are good for 5000 pounds, so even if the water weighs several hundred pounds we’d never notice.
The most important factor in engine performance is the density of the air, which is driven by temperature and altitude. You get more power on cold days and at low altitude, less on hot days and high altitudes. (Which is why Denver has long runways)
There is a decision tree when planning a takeoff, and extended stopping distances due to a wet runway sometimes pushes you to use a higher power setting which is a bit less efficient. So the answer to your question is maybe a little, sometimes, but not in the way you think.
Total guess but I imagine the increase in relative humidity impacts the combustion efficiency of the engine as well
Fun fact. In the days before high-bypass turbofan engines, water was deliberately injected into jet engines to cool down the combustion chamber and increase thrust: https://en.wikipedia.org/wiki/Water_injection_(engine)#Use_in_aircraft
Water injection is still used in some industrial gas turbines as a control for emissions, along with a modest increase in power. Steam injection is also used in some situations.
Damn what a rabbit hole and TIL
