Car sold, what now?
The sales contract ends neither your vehicle tax nor your insurance. As long as the vehicle is registered to you, you are liable as the keeper.
The check costs nothing and you reveal nothing during it.
Three routes, and only one is yours to decide
The buyer re-registers
The usual route. Ask for proof, and set a deadline.
You deregister
If the buyer does not deliver or the vehicle stays put. That is your call.
Notify the sale
Report the sale to your registration office. That protects you, but it does not replace the deregistration.
Important: to deregister you need both number plates and the original Zulassungsbescheinigung Teil I. If those are already with the buyer, you can no longer deregister.
What happens if you do nothing
The vehicle tax keeps running in your name, and so does your insurance. If the buyer drives uninsured, that can come back to you. Fines from traffic offences land with you first, because the register still shows you as the keeper.
First check whether it works for you
The check costs nothing and you reveal nothing during it.
Check for free whether it works