FlightCar is a business that uses another entity's property and services without permission and for profit,but without remuneration. This is not at all comparable to a private individual doing a friend a favor for free. I would imagine that the airport would start with a cease-and-desist letter, perhaps along with an offer to come to to some sort of revenue-sharing agreement for continued access. Failing a voluntary resolution, a lawsuit might follow.

To address your last point, let's use another example. Football stadiums are public facilities often owned by local municipalities. But that doesn't mean I can go in and start selling hot dogs.