How conservation of energy looks like in the moving frame? It is obvious that a car accelerates by converting its chemical energy in the fuel to produce kinetic energy to accelerate itself.
If the energy is lost into friction and heat, the car will slow down.
But from the point of view of the car, it always appears to be stationary, which means its kinetic energy is zero. So what is the thing that the fuel actually does in the car frame?
 A: From the car's point of view, there is a fictitious force, which does work. Therefore, the car sees extra potential energy. You can think of the fictitious force as weak constant gravity. Similar to how you can see gravitational potential energy, in the car's perspective, there is the fictitious force's potential energy.
Edit: Because I didn't actually answer the question. What's happening to the energy in the fuel in the car's perspective is that it's being converted into kinetic energy. The car has to push things backwards in order to move forward. What the car sees is that it's using the fuel in order to push things twice as strong backwards. The reason why the car isn't moving forward in the car's perspective is because momentum isn't necessary conserved because there is a force in the non inertial frame of reference.
A: Energy is not invariant across reference frames. From the car's frame you would correctly calculate the car's kinetic energy to be 0. The earth, on the other hand, would have massive kinetic energy.
The fuel does work against the pistons in the engine. The energy is lost in the same places as before.
A: Let' say there are two earths, on one of those earths a car is accelerating.
Now the car says that the kinetic energy of the earth that the car is not pushing increases by a huge amount, and the kinetic energy of the earth that the the car is pushing increases by the same huge amount plus the energy of the burned fuel. Or it increases by the same huge amount + force * distance.
