Faster Small Tires vs Slower Large Tires We have two tires with radius r and R respectively. They are connected to two almost identical vehicles with same net mass.
"Almost", because Gear ratios are adjusted such that At equal Engine RPMs, both vehicles have same road speed, on paper at least.
So, does this have any effect on the offroad performance, hauling performance, fuel efficiency, and power output of the vehicle?
What else effect could wheel size have?
Would the effect be different for different vehicle builds, like semi-trucks, minivans, bikes, muscle cars, and race cars?
This question is just a thought experiment, so contruction cost, logistics, looks of vehicle etc do not need to be considered.
r is small radius and R is large radius
 A: As per rolling resistance law, ratio of engine work done opposing rolling resistance force negative work ,in case big and small tires is :
$$ \frac {W_{_R}}{W_r} = \sqrt {\frac {r}{R}} ~~~~~~~~~~~~~~~~(1)$$,
assuming same sinkage depth in both cases. $W_{_R}, W_r$ - rolling resistance negative work done of big and small tires respectively.
If sinkage depth $z$ depends on tire size (which should be, because bigger tires should have higher contact area with the ground, this in turn should reduce car pressure to the ground, and consequently - sinkage depth), then per coefficient of rolling resistance,
$$ C_{rr}={\sqrt {\frac {z(d)}{d}}} $$
where $d$ is wheel diameter, relationship of work done ratio against rolling resistance force given in (1) can be more complex.
In any case, as per (1) it is expected that a car with bigger tires will be more inertial, and thus will consume less fuel for keeping same speed. Conclusion, bigger tires will make car more fuel effective.
A: Long distance trucks use large tires in part because of rolling resistance,  as @AgniusVasiliauskas pointed out.
But also a large tire has a more gentle curvature. This means a larger contact patch. This means more traction for a race car. An off road car won't sink as deeply in sand or mud because of reduced pressure under the tire. This also means reduced tread wear.
When encountering a bump,  large and small tires are deflected upward the same distance. But the large tire feels the bump sooner. The deflection takes place over a longer time interval and is gentler.
