Bouyancy is simply about relative density - the density of an ambient medium and the density of an object within that medium.

In the scenario described, there is a fixed quantity of air in the tires. To do their normal job, the tires will be inflated to (perhaps) 30 psi or so over ambient air pressure (out of the water). Released from the tires, but still contained in a bag, you’ll still have the same quantity of air (mass will be the same), but at a lower pressure it will occupy a larger volume. The total system (car, tires, and bag) will not change in mass, but will displace more water, so will have lower density, which means greater bouyancy.

Movie science must always yield to drama and the needs of the story ; in the movies, there’s certain to be more than enough air in the tires to float the car ; in the real world, probably not. It would have the effect depicted (would increase bouyancy), but almost certainly not enough to achieve the result appearing on screen.