Besides the nerve, what else could be damaged/dysfunctional if a cranial nerve reflex test does not elicit the?

Well, entire parts of the brain could be damaged. For instance in a brainstem bleed or stroke, parts of the brainstem will be nonfunctional (which could have other effects, like decreased respirations, abnormal posturing of the body, etc) in addition to the damaging the cranial nerve than originates from that brainstem area.

What particular reflex did you have in mind? Go through the reflexes- each has an afferent arc and an efferent arc. Damage in either, whether it be the nerve or the nucleus, can affect the reflex.