My water bottle seems to be under extra pressure when I open it inflight. Are the tires under pressure too, and do they expand much?
Your water bottle experiences the result of a depressurization of its exterior as the cabin pressure decreases during climb. The interior air seems to be of a higher pressure. Of course, that is all going on inside the cabin where the cabin pressure is still about 8,000 feet at cruising altitude. However, the tires are never in a pressurized environment. When at 41,000 feet, they are nearer to space than on the ground in terms of pressure. ("Hey, what about the Space Shuttle tires?" Same as airliner tires. In fact, I think they use the same tires we do.) Filled to over 100 pounds per square inch on the ground, if they weren't strong, they'd explode for sure at altitude. The key is how they are made.
Car tires have a couple of belts to add strength and durability to the tire. Otherwise, side loads and direct loads would pop the tire or push it off the wheel rim. The rubber used for tread has little strength. Some supporting structure must take the load of the vehicle. This structure is called the "carcass" of the tire. Light truck and SUV tires have 4 plies to add the strength needed to handle the stress they encounter holding up a bigger vehicle. To handle the weight of a jetliner like the 737, the tires have 28 plies! These tires are so strong they can handle the punishment of landing and braking as well as the strain under pressure at cruise altitude.
With the Aramid fiber used in tires having more strength than steel (pound for pound), I doubt the tires expand too much at altitude. The tire "carcass" is so strong that it allows the airline to retread the tire several times to get better life out of them and keep the cost much lower than buying new tires all the time. You see the importance of this when you watch an airplane land because the "smoke" you see arising from the tires is vaporized tire tread. It doesn't take long before there is no tread left (and tires are inspected every time a Pilot does a walk around), but again, the steel-like carcass of the tires is what supports them.