Is there any theological reason that Eastern Orthodox clergy tend to maintain long beards? I have minimal familiarity with the Eastern Orthodox tradition, but I can typically guess by seeing a photo that a person is an Eastern Orthodox clergyman.
It's a tradition derived from two sources: first of all, Jesus had long beard - second, from Leviticus 19:27 "You shall not round off the hair on your temples or mar the edges of your beard"

However, it's not a prescription for ortodox clergy to do that: it's only a (widely observed) tradition
Note that catholic clergy usually not maintain long beards, for a different tradition: Romans used to shave their beards, and to consider men with long beards as uncivilized (barbarians).
Anyway, there is no prescription about beards (exept for some monks)