Yes, you may be charged an additional fee to travel to another country with the car. This fee covers additional coverage, taxes, and so forth.
✏️ Make sure you get permission beforehand; otherwise, you may be charged a penalty and lose any coverage your car has, thus becoming fully liable for any damage to or loss of the vehicle.
The cross-border fee won’t cover tolls, vignettes, or road taxes in the country you're planning to cross into.