Not in general. Think of it like this: when the car drives off the cliff, at some point it has a portion of it falling and consequently rotating due to torque exerted by gravity while the other end remains at its original height due to normal forces.
As a result, the car will tend to have some sort of rotation as it falls. Thus, the height of the cliff would play a role insofar that it must match up with the distances required for full revolutions of the car's rotation.
The physics here is simple, but it suffices to show that it is certainly not always the case that the car falls upright.
Moreover, the speed at which the car is being driven ought not to matter under the assumption that the cliff is flat. Indeed, mass effects could be significant given drag forces, but they're not necessary to show that the claim doesn't hold in general.