I paid off my car loan --- now what?

If it's a lease, then you get an entry in the grey card that says you cannot transfer the car to someone else. When everything is paid out, the lessor will notify the STVA and you can update your grey card so that entry is removed.

There are no car loans with lien in Switzerland. You get a personal loan, unsecured, and you can use it to get a car or whatever you like. When that's paid, you just don't owe money to anyone - nothing else changes.

So if you have an entry in the grey card, you can get a new one without an entry.

If you don't have an entry in the grey card, you don't do anything.