Actually, such a state exists!
The Vatican is a linear descendant of the Roman Empire, based from the Bishop of Rome. Although a micro-state in every definition of the word, the Vatican really can claim to be the distant successor to the Roman Empire, with an army of faithful and influential, not coercive, powers. Besides, the heavy usage of Latin, etc, etc, etc...
Oh, and its located in Rome itself.Wow, I'm on fire today.
The problem is that Rome is as much Geography as Political. Byzantium was definitely an outgrowth of "Rome" but it was very clearly a whole different player for a far longer time. Byzantium is much more "Greece" than "Rome" if it must be said.
It might be borderline ASB, but what about a Roman Noble Family that survived to the modern day? You could claim a chain of continuity for two thousand years. That would be very interesting--but considering how many royal families can and do date back from the middle ages, I suppose that this is possible in one way or another--perhaps the Family moved to Byzantium and then Regained Prominence in 11th Century Italy?