If you start with Newton's 2nd law:
F = m * a
then you see that for a larger mass (m) to be accelerated, you need a larger force (F). There's only one way to get accelerating force in a car (at least on flat ground!) and that's to burn more gasoline. So a car with a full tank of gas does require slightly more gas to accelerate, so your overall gas mileage is slightly lower. On the other hand, the mass of a full tank of gasoline (40-80 kg) is no more than the mass of a person (50-80 kg), so it's not a big difference. Also, if you have to drive to a gas station twice as often, you'll be wasting much more gas than you saved by only driving on a half-tank.

A similar question was answered earlier this year on Car Talk, a National Public Radio show:

The gist of their answer was that yes, a full tank weighs more than a less-than-full tank, so theoretically the full tank would reduce gas mileage. They pointed out that the weight difference is minimal because cars are already so heavy, so how full the gas tank is doesn't really make a difference for gas mileage at all. It was even suggested one might waste gas driving around to gas stations more often if they only wanted to keep their tank half full.

However, here's that theoretical bit related to reducing gas mileage. Newton's second law is F = ma (force equals mass times acceleration). Because a full tank of gasoline does increase the mass of a car (albeit very slightly), the amount of force required to move the car at the same acceleration is greater. Likewise, the work done on the car (I mean in terms of physics, not sending it to the mechanic) also increases when the mass of the car increases.

The short answer is "yes, but so small you would never notice", but the physics are quite a bit more complicated than the simple relationship between force, distance, and energy.

Newton's second law, if I remember right, is the definition of force, i.e. mass times acceleration. Work (energy) is the integral of force over distance (or force times distance, if force is constant over time). However, the force that opposes your automobile's engine is not the weight of the car, but the friction generated by the car and by the engine as it rolls along. And this friction is not simply proportional to the weight of the car, since a car does not slide along a surface but instead rolls on wheels, and there also is internal friction, since the entire car does not roll: the wheels are attached to an axle, which is in turn attached via a transmission to the engine. Most of the friction generated when the car is in gear is actually generated in the engine and transmission; this is why a car takes a whole lot longer to roll to a stop if you shift it into neutral, because then the transmission is no longer involved. Now, ultimately, all of this friction is due to the weight of the car, and having a full tank of gas does make the car heavier. Still, the weight of twenty gallons of gasoline is negligible compared to the weight of the car itself, which is usually a ton or more.