Which aircraft require that the vertical position of the centre of gravity is checked?
In many aircraft the vertical (Z axis) placement is of little impact on weight distribution.  However, in some aircraft it affects maneuverability.  So for an aircraft like a C-5 it is noticeable, yet a C-5 is not considered an aircraft where vertical CG would be a factor, as the nature of the operations are, well, cargo hauling.  In a helicopter, which tends to have a higher (Z-axis) CG with the top placement of the engine and transmission, the considerations are greater. It doesn't impact loading in a normal sense, but there is a noticeable impact in handling particularly when roll moments are involved. 

Stated perhaps more simply, static stability is normally bounded by longitudinal and to a lesser degree latitudinal forces. However, a higher CG will impact dynamic stability where the consideration is not primarily limited to wing (rotor) and airfoil loading.

So the simplest answer is that it does make a difference, which is more than just measurable in some applications, but normally the difference is of small consequence and only a tertiary consideration.