Typing “speed of light headlights” into a well known search engine provided 296000 answers.

Nothing.

If the car is going at the speed of light then the electrons in the wires would have to break the speed of light to get to the head lamps. Since this cannot happen, the lights will not work.

As far as "the car" is concerned somthing else (you) is traveling at the speed of light and absolutley nothing inconsistant with the laws of physics occurs.  Re: the first postulate of relativity.

questor says...

"As far as "the car" is concerned something else (you) is traveling at the speed of light and [in that frame of reference] absolutely nothing inconsistent with the laws of physics occurs." 

If you were tailgating the car ahead at the speed of light, the headlights would shine on it's bumper.

(Your frame of reference is that you left earth and have accelerated to the speed of light... using a source of infinite energy supply.)

The Second Postulate of the Special Theory of Relativity The second postulate of the special theory of relativity is quite interesting and unexpected because of what it says about frames of reference. The postulate is: The speed of light is measured as constant in all frames of reference. This can really be described as the first postulate in different clothes. If the laws of physics apply equally to all frames of reference, then light (electromagnetic radiation) must travel at the same speed regardless of the frame. This is required for the laws of electrodynamics to apply equally for all frames.