Firstly I'll precise that I never worked on mission design. I worked on aerospace long ago, but not about that.
To my knowledge there's no fuel reduction due to variable payload. This is the same rocket, it's designed for a range of payload mass for a particular destination orbit (LEO, polar orbit, GTO...). Not filling completely the tanks could result in a different behaviour: the rocket is lighter but the engines provide the same thrust, so it accelerates stronger, so it's legitimate to wonder how it would behave, when passing Max Q typically. The launcher could also behave differently regarding vibrations, or even the hardware could behave differently. The first Ariane 5 is a good example of that: the inertial central failed because it wasn't designed for such a strong acceleration (the context was different however: the inertial central was the same as for the less powerful Ariane 4, it's been reused for Ariane 5 for saving purposes).
In practice, launchers are adapted by having different configurations (different number of boosters, or different second stage). You couldn't really partially fill a solid rocket booster anyway: it's a mixture that is molded into the casing with a particular shape, and the thrust curve will depend on that shape (the boosters' thrust is not constant over time). Changing the quantity of rocket fuel will change all those parameters. For cryogenic engines, you can stop them whenever you want anyway, and the excess fuel is vented once the mission is over. This is stage passivation, to avoid an explosion that would spread debris in orbit.
Also, on an economical point of view that would make no sense: fuel is only a small fraction of a launcher cost. The most expensive part is by far the engine itself, so saving a little fuel is probably not worth it. This argument loses of its relevance if you consider reusability though: since the engines are reused over several flights and only the fuel remains a consumable, their cost has to be spread over se veral flights, so the fuel represents a higher cost in proportion.