"Rotationally inactive" is probably about the rotational spectrum of the molecule
I'm assuming the (unsaid) context of the question is that it is about rotational spectroscopy as this is where the term "rotationally inactive" makes sense.
Rotational spectroscopy–like its better known cousin infra-red spectroscopy–measures absorptions of electromagnetic wavelengths corresponding to transitions in molecules. IR spectra mostly involve vibrations in molecules that have energies corresponding to wavelengths of IR light. Rotational spectroscopy (usually gas phase, as molecular rotations in solids are not usually easy) measures rotational modes in molecules in a similar way (strictly speaking for both techniques absorptions happen when the wavelength/energy of the radiation matches a transition between two rotational or vibrational modes).
But for an absorption to happen the radiation has to have a mechanism to interact with the molecule. For both vibrational and rotational spectroscopy, this can only happen if the molecule has a dipole moment (or, strictly, the relevant vibration or rotation has a changing dipole). While different vibrational models in the same molecule can be "active" (eg the asymmetric stretch in carbon dioxide) or "inactive" (the symmetric stretch in carbon dioxide), in pure rotations there is no way for a molecule without a dipole to have an "active" rotational spectrum as rotation doesn't change the dipole. So only molecules sufficiently asymmetric to have a dipole moment can interact with the radiation. Conformer III of hydrogen peroxide is the only conformation with no net dipole so is the only one which is "rotationally inactive".
The terminology is confusing outside this context as it could be taken to mean something to do with the interconversion of the different conformers by rotation around the central O-O bond (which is likely fairly easy).