The short answer is no.
The long answer is probably, but you'd need to entirely replace most of the control logic for the CD player.
It's true that the actual mechanical assembly used for reading a plain-old audio CD should be entirely capable of reading a CD with MP3s burned onto it.
The decoding logic is another story. Simple audio CDs basially have sequential DAC vaules encoded on them. The "decoding" basically involves reading the sequential values, and sticking them on the output DAC (there is a bit more to it, generally related to buffering so if you hit a bump, it doesn't interrupt the playback, but that's the gist).
Decoding MP3s is significantly more involved. There is very little likelyhood that the electronics in a CD player could be adapted to do MP3 decoding. Even if they were capable (it actually takes a pretty significant amount of processing power to decode an MP3. You couldn't do it on an arduino, for example), you would need to be extremely familiar with the embedded device they use to run the CD player.
This is all assuming that the device used to play back audio CDs isn't a specialty chip designed just for CD playback, and is instead a reprogrammable microcontroller. If the CD player uses a purpose-specific controller, the only way to make it play MP3s would be to entirely replace the control electronics with your own.
Realistically, if the problem is that when you turn off the car, your MP3 player keeps playing, there is probably a simpler solution. You just need a mechanism that automatically stops your MP3 player when the car is shut off.
Many MP3 players have line-out outputs, that are only active when they are connected to a charger. If you have a charger that only outputs power when the car is on, that would solve the problem quite easily.
Realistically, any further solution is probably going to be specific to your MP3 player, and would probably warrant another question.