It did play a role, but I don't think we can quantify just how important it was, mostly because of the lack of sources and various problems with the sources we do have. In my previous answers (on the Arab Conquests generally and on the conquest of Egypt/North Africa), I mentioned that the miaphysites (the modern scholarly term for monophysites) were most likely not keen to fight to the end for the Roman Empire, but I didn't clarify whether they would have fought against the Romans. This is because there is no evidence of miaphysite Christians doing this and later miaphysite historians, our main source of information for this period, may have covered this up even if it happened.

In Egypt for instance our most important source is John of Nikiu's Chronicle, which was written a generation after the conquest. He was generally quite positive to the Muslims and condemned the Chalcedonian patriarch governing Egypt during the war, but at no point did he say anything about active co-operation. He did mention several apostate Roman officials who helped their new rulers, but there is no way for us to figure out if they were genuine apostates or just collaborators. I'm less familiar with the Syrian sources, but again I'm reasonably sure that we don't have any sources describing 'active' collaboration with the Arabs. Philip Wood's We Have no King but Christ. Christian Political Thought in Greater Syria on the eve of the Arab Conquest (2010) covered the development of a Syrian identity in this period, so you might find that interesting. Even so however he made the point that this identity did not mean that the miaphysites were disillusioned with the empire - they hungered for an orthodox empire (according to their definition of course), but that did not mean that they active hindered the empire's actions in the seventh century. Another example would be the Armenians, who were also miaphysites and they did abandon their Roman overlords when the Arabs invaded, but the Armenians (and other Caucasian kingdoms on the fringe of the empire) regularly switched sides in the wars of the sixth century to secure their autonomy, so to what extent was this due to their political interests or religious faith? Probably a mixture of both, but I don't think there's a clear-cut answer about one or the other's importance.

More generally, we need also to note the Roman Near East had already collapsed before in the 610s against the Persians (who did favour the miaphysites apparently), so the fact that further Roman defeats in the 630s led to mass capitulation isn't that unexpected. The Arabs weren't there to destroy Christianity, so all Christians had very good reasons to negotiate with their invaders once there were no tangible Roman forces nearby - the same had happened throughout the wars between the Romans and the Persians for centuries. There were other groups in the empire who may not have been keen on fighting to defend the empire, such as the many Arabs who already lived in the empire and they most likely took part in the conquest as well, as the armies streaming out of Arabia preached something that was especially appealing to Arabs. There were also internal social divisions, such as between the Blues and Greens (whose riots caused a massive crisis already in 608), and of social conflict between the rich and the poor, a problem that had plagued the empire since its inception.

It is also worth pointing out that the Pact of Umar may not be a genuine document from the seventh century. Still, Islam's toleration of other Abrahamic faiths is well attested to, so the spirit of the document is generally true. I can't comment on what it means by the 'parting of the hair' comment I'm afraid, though I'd suggest it would have to do with distinguishing between Christians and Muslims (though I'd also add that some scholars might not think there's such a stark distinction in the first century AH, as people like Donner would argue that Christians were very much part of Muhammad's movement, so why would Umar attempt to separate the two communities?).