It certainly does. To fly, an airplane has to push air down, it creates wing tip vortices, these vortices are wasted energy. The heavier the plane, the more air it has to push, the stronger the vortices and the more energy is wasted.

Cars push on the ground, and the ground gives back all the force applied to it, so no losses here, planes that fly very close to the ground can also take advantage of the ground effect (that's the name) for more efficient lift.

To how much extent, I don't know, but one thing for sure, manufacturers strive to make their planes as light as possible.

As a side note, heavier gliders actually fly a bit better (faster) but unlike powered airplanes, they don't lift themselves.

Edit: added ground effect

Not as much as you might think. A huge amount of energy is required to gain the altitude for long distance flight. Cruising much less so.

The way that airplane wings work allows planes to sort of float on a cushion of air, so modern planes, for example a 747 at cruising speed, have a lift to drag ratio of 17. So one pound of thrust gives you 17 pounds of lift.