I think that’s a load of bunk. The drag caused by a pickup bed with the tailgate closed is negligible.
I used to have an S-10 that came with a bed cap. The thing was really heavy, at least a few hundred pounds. The truck felt a bit sluggish with it on, but ran much easier with it off and got better mileage without it (not a lot, mind you).
This sounds like the sort of thing a tonneau cover/bed cap manufacturer would say. Any difference would be miniscule, and fuel savings would be immeasurably small, if present at all.