I guess technically it does.
But the only reason I know of that it should is the relativistic increase
of mass with speed ... that's why we never notice the increase at
everyday speeds.
The effect gets larger at higher speeds. For example, if the car is
cruising through the neighborhood at 6.71 million miles per hour
(1% the speed of light), then its mass, and therefore its weight,
is 0.005% more than when it's sitting still at a red light.
Now, if the driver were to put the pedal to the metal and open 'er up
to 10% the speed of light, then the car's mass (and the driver's mass
too) would increase to a whopping 0.5% more than its 'rest mass'.
So you would definitely have to say that the vehicle does get heavier
as it speeds up.
