Volume nor pressure change at all. Any air that was in the section of tire that is pushed against the ground gets relocated through the rest of the tire. Pressure does not change, nor does volume of air, unless you have a leak.

Basically speaking, the tire inflates outward from the rim, so the tire does nothing to support the rim other than give a cushion to protect from bending. The tire is merely to add cushion and traction. If you didnt have a tire on the rim, it wouldnt bend just from your weight, but it would bend when you hit bumps because there is no spring back to change the direction of pressure against the rim.

Tires dont structurally support rims at all. In fact, the rim supports the tire and keeps it from raising off the to of the wheel when weight is rolling on them.