The Mini 4WD's tires aren't full of air, and it can run. Also, the tank doesn't have tires with air. So, the question is: why do real cars on the road need to be filled with air? What is the idea behind it?
Answer
You don't want to lose energy – not only because of energy efficiency but mainly because the desire to achieve high speeds and reduce the deterioration of the wheels – when the wheels are changing their shape due to the pressure caused by the weight of the vehicle.
If you want to squeeze the wheels with rubber by a centimeter, you need a substantially greater force if the tires are filled with air whose pressure is many atmospheres than if you have "just the rubber". So inflated wheels at 4 atmospheres guarantee that the wheels are more rigid and the ride is smoother.
This advantage becomes more important at higher speeds because the wheel could get deformed many times a second at higher speeds. And it becomes more important for greater vehicles (relatively to toys) because their mass grows as $R^3$ with the radius $R$ while the area of the wheel is expected to grow as $R^2$ only. So the pressure – force per unit area of wheels – becomes greater for greater vehicles. Not an issue for toys but large vehicles need to have wheels that are resilient under much higher pressures.
On the contrary, the need for high pressure in the wheels decreases if the area of the wheels is large (the wheels are "wide") and if the car is relatively light. So formula one cars only use about 1 atmospheres in the wheels. Also, mountain bikes (thick wheels, bumpy roads, expected low speeds) often have 2 atmospheres only while racing bikes (thin wheels, smooth roads, expected high speeds) may be pumped to 15 atmospheres.
Tanks are a different issue because the rubber is relatively very thin (relatively to "whole wheels made out of rubber") and it's the metal "right beneath" the rubber that is resilient and doesn't resist the pressure. Moreover, their speed is lower and they are going through a non-uniform terrain in which the adjustment of the shape of wheels or their counterparts may be a good thing.
Another question, kind of opposite to the original one, would be why the wheels are not just made out of metals. Well, that would be an uneasy ride and the metal would be damaged very quickly, too. One needs some "spare room" so that the tires' thickness may change by a few centimeters if it's really needed. On the other hand, one doesn't want the tires to change easily. A layer of the air at high pressure is a great and simple answer to both conditions.
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