بسم الله الرحمن الرحيم
For a start, there were wars between Roman Empire which conquered Syria and Persian Empire which conquered Iraq. The rivalry must have started from that or maybe before it.
Salam brother i'm not sure about this. This is the map of the extent of the Byzantine Empire:
http://www.regentsprep.org/Regents/global/themes/power/images/byzant.gifYou see that it just covers Jordan and Palestine. Rather modern Syria, like Iraq, was controlled by the Sassanian empire:
http://upload.wikimedia.org/wikipedia/en/4/41/Sasanian_Empire_cca._620_A.D.png(this was in 620 c.e)
However as i mentioned to brother Hani, there was the ~40 year Harb al-Basus between Bani Taghlib and ٍShayban. This is a map of their territories:
http://upload.wikimedia.org/wikipedia/commons/thumb/6/68/Al-Jazira-ar.svg/1586px-Al-Jazira-ar.svg.pngِAnd not only was this war fough between these two tribes, but also allied tribes (kinda like how WWI blew up), as evidenced by poems like this:
صفحنا عن بني ذهل *** وقلنا القوم إخوان
عسى الأيام أن ترجـ *** ـع قوما كالذي كانوا
فلما صرح الشر *** وأمسى وهو عريان
ولم يبق سوى العدوان *** دناهم كما دانوا
However this was like a hundred years before the birth of the Prophet(pbuh) and we know that these tribes united to fight off a Persian attack and actually won in the Battle of Dhi Qaar (and some narrations narrate that this victory was by the blessing of the Prophet(pbuh) بي نُصروا even though they weren't Muslim)
(great lecture on both of these battles by a great historian here:
حرب البسوس - حرب ذي قار)
So i'm really not sure whether that enmity still stayed.
But tbh what is apparent to me that the cause for the schism was Mu'awiya moving to Shaam and setting up his own state there and going to war against 'Ali(as) and by extension the people of Kufa.
Also they were just simply different people. The Islamic Empire was divided up in an interesting way. The people of Madinah were known for 'ilm and the hadiths of the Prophet(pbuh), the people of Shaam were known for jihad, and the people of Iraq were known for being worshipers ('ubbaad, zuhhaad)
i really don't think ancient enmity between the Byzantines and the Persians had much to do with it. Are you aware of any poetry or historical records that say this?
But i really don't know and Allah(swt) knows best