it will affect gearing obviously but a lot of thetime attack cars etc will run specific gear ratios (and final drive) to suit the needs of the track, which will take into account the diameter of the wheels
The diameter of the wheels is usually dictated by suspension design, cooling, braking etc etc
as for road cars, a more effected way of changing the gearing is to change the final drive (which will be more dramatic than a wheel and tyre change) which then allows you to select wheel size based on neccersary tire size and width, brake size, aesthetics etc etc
in short yes you can, but there are better ways to do it