Hello... does anybody know if it is possible to run the Air Video OSX Server on
the AppleTV to stream videos from the attached USB storage to the iPhone or
iPad?

Requires 10.5

AppleTV is based on OS 10.4 (with a few things backported from 10.5)

Apps that require 10.5 won't run on 10.4 or the atv at all.

Even if you could get past that limitation the CPU of the atv does not gave the power to transcode on the fly - so you could only stream videos already in mp4/iPhone friendly format.