I had this bug happen to me and a couple of friends both in modded and vanilla servers. What would happen is, I would be driving a fully functional car for a while. Then I would get off without turning off the engine, fight some zombies or something. And then, when I go back to drive the car again, the car would be in sort of a "frozen" state. The engine light would be on but the gas and rpm would be displaying zero. And then, when I press "W" to accelerate, I would get engine rev noise but no movement. Also, when this bug is happening, I cannot remove the battery (I'm sure every other part would be the same but I haven't tested it). The only fix for when this happens is, if another player who is not experiencing the bug gets in my car, and turns off the engine for me (because I am unable to turn off the engine). And only then, can I turn the engine on again and drive normally again.
Another iteration of this bug is, I get on a car and that car is already bugged and I get the same symptoms except that when I accelerate, there is no engine sound.
UPDATE: I also found out that I can turn on the lights, lock the doors, but cannot turn on the heater because it is registering as my engine is off, however, the engine light is on and the car is making revving sounds.
Here is a video example
