Less fuel onboard is more efficient because it is less weight..
You can see this method used for race qualifying and can also see laptimes pick up as fuel is used up in some classes where weight is critical..
But in just a daily driver car, filling it up every time and running it almost out may be the most efficient not having to go out of your way to gas stations so often.. With the high relative weight of an average car it is also likely negligible..
Well now that I've thought about it, you're right. I remember a trivia where a bridge has a limit of 1 ton and its distance is around let us say a 100 km and your truck weighs 1 ton. And when you're halfway across the bridge, 50km, a bird suddenly sits on the roof, adding more weight to the vehicle. Will the bridge collapse??