You have a rivalry with your rivals -- generally -- because your school has some kind of culture conflict with the other side. And often the conflict is local or intra-state, which brings it closer to home. See:

AUB vs. ALA
TX vs. TAMU
GT vs. UGA
CLEM vs. SC
etc, etc, etc.

But I don't think VT folks have too much culture conflict with GT folks -- there are more similarities than differences.

The big problem VT folks have with GT is the actual product on the field -- that damn option game. Anybody who plays GT (whether GT is good that year or not!) has to disrupt EVERYTHING about game prep and game concepts, at least for that week. GT week becomes a giant mess inserted to the middle of every season. It's like coming down with the flu once that winter weather change comes around. Most times you survive it, sometimes you don't -- but you are always sick, without fail.

Everybody hates that, and just about everybody would like to see GT off their schedule as a result. But that doesn't mean we actually hate you guys the way Michigan hates OSU, or even the way VT hates UVA (which is much more gentlemanly).

So I would say GT and VT don't have a rivalry -- from our perspective it's more like a disease.