I don't think that a "no Islam" scenario butterflies Arab expansionism. There was an overpopulation problem within the Arab community at the time, and they had to go somewhere. I would not be surprised if some other Arab takes up the mantle of conquering large swathes of the Middle East for some reason... perhaps driven by a desire to Christianize everyone, perhaps not.
If in the name of Christianity, it would make little sense to go trumping about North Africa. Greater Iran and the Arabian Peninsula are likely to be the center of the Arab conquests in this case. However, what comes of the Arab conquests is likely to be very different. One of the primary reasons that with conquest came acculturation into Arab society is because the conquests were in the name of a highly nationalistic religion (at the time at least) called Islam, which mandated that Qur'an not be translated outside of the original Arabic. The religion in and of itself had a lot of Arabic cultural norms of the period tied up in it as well. Without a religion that is in a number of ways intensely Arab-centric, Arab conquests are likely to look a lot like the Germanic conquests in Europe. Tens of thousands of Arabs on the move, perhaps they conquer Iran and Mesopotamia (in the "in the name of Christianity" scenario), but the conquerors eventually absorb into the communities of the conquered, not only linguistically, but culturally as well... for the most part.
That means that yes, you may have a surviving Assyrian Fertile Crescent, and that Iranian dynasties may continue to be the center-piece of Middle-Eastern politics for awhile, but in a somewhat changed form. You would probably have Aryanized Arabic dynasties in Iran and Iraq for awhile, until something local supplants them, before the Turks/Mongols come swarming in, if that ever happens.
I myself have always thought that the idea of a Hungarian Anatolia via a Magyar invasion of the Middle East to be a very interesting prospect... but I digress.
In a non-religious context, where the Arabs are moving about the Middle-East, conquering as they go, NOT in the name of any kind of god, the situation would be similar, although they may just enter North Africa via Egypt as well as going into Greater Iran and even Anatolia. They will probably conquer, pillage, and rape quite a bit, but in THIS scenario, they will probably acculturate a lot quicker. The reason being that they are not coming to Iran or Egypt or Anatolia to teach heathens a lesson - they are coming for land, and for wealth. This situation might be more analogous to the Normans, who acculturated REALLY quickly (when you compare them to the Visigothic aristocracy of Spain, which may have spoken Visigothic amongst themselves as late as the Moorish Conquest). Arab names will be popular within the nobility and maybe even the people, but their meanings obscure or forgotten as the Arabs will speak Assyrian/Persian/Greek/Coptic/Kurdish/Berber. There might be no Moorish Conquest of Spain or Sicily, but if there is, the Arabs will again, be absorbed pretty quickly. Again, they're not necessarily here to change the lives of the local people, but they're here to change their own lives.
What are the long term effects of this?
Well, first of all, you see some more linguistic diversity in the Middle East, probably. I imagine Old South Arabian languages would still be spoken today, even if they were just minority languages. BUT, more importantly, Arabic will not enjoy a rich literary tradition in many places outside of the Arab homeland... probably not at all. Greek, Assyrian, Persian, and Romance dialects have much more extensive literary traditions going back thousands of years. Why should that change just cuz some Arab barbarians came in and burned some houses down and killed some people? Also, African Romance languages could possibly survive into the modern period in this scenario as well. There are historical references to African Romance languages being spoken as late as the 1100s. Without Islam to force everyone to speak Arabic in order to read the Qur'an (that includes Berbers), African Romance languages will likely continue to be spoken alongside Greek and Berber languages. Morocco, Algeria, and large swathes of Libya and Tunisia might be entirely Berber-speaking... imagine that.
Another consequence that I think anyone who ignores is kidding themselves, is that religion may cease to play a significant role in politics much earlier on.
There were already Christians/Buddhists in Iran at the time of Arab conquest, and without the threat of Islam to contrast itself to AFTER Europe had been Christianized, Christianity could lose a lot of its momentum that kept it at the center of politics for the better part of 1500 years. This could happen ESPECIALLY if the Middle East is Christianized into Greater Iran (the first scenario). You will probably have some bickering among the patriarchs about who has the bigger the dick, but it is unlikely to cause the kinds of serious conflicts that we saw OTL (The Crusades).
In such a scenario, cultural exchange between the Middle East and Europe is likely to be more pronounced, since they are not as separated on the grounds of religion. Yes, Western Europeans might believe that the Patriarch of Rome is the only patriarch worth answering to and that people who answer to the Patriarch of Alexandria or the Patriarch of Ctesiphon are heretics, but heretics are not heathens. Heretics are people who have a different interpretation of the same message you believe in. You can have a conversation with a heretic about a lot of things that you have in common in terms of religion because there is a basic groundwork that has already been laid. Heathens don't have that basic groundwork. You have to start with THEM from square one.
If you're religious, think about a Mormon trying to talk to a Catholic. Catholicism and Mormonism are VERY different in their institutional set up, but many of the core beliefs are actually very similar. At least both of them believe that Jesus Christ is the son of god and they both read the New Testament (albeit different versions). Then think about trying to talk to a Muslim. Do you even know who Muhammad is, or what he did, and why he is important to Muslims? Do they even believe in Jesus?
So people in the Middle East are going to be a lot more relatable to people in Europe on these grounds. Barring some kind of nationalistic movement in Europe (like existed in Greece during the time of Alexander), Europe and the Middle-East will trade and exchange ideas easier. Indian/Chinese maths are likely to come to Europe earlier, Arabs probably will stop circumcising, etc.
This doesn't mean that no Islam is the key to a golden age of enlightenment, though, cuz we have the flip side... what if religion is NOT a factor in the Arab conquests?
I think that religious diversity in the Middle East is quite probable in this case. Mandaeism, Manichaeism, Christianity, Yazdanism, Buddhism, Zoroastrianism, and MAYBE Judaism (Judaism has some nationalistic tendencies it needs to overcome first) will all rise and fall, but Christians will surely be able to contrast themselves against one of or all of them. It all depends on the religion of the area of affluence, which is still likely to be Mesopotamia, I think. Are Assyrians Christians, are the Mandaens, Manichaeans, Buddhists, Zoroastrians? They'll probably be Christians, but we can't say for sure. Whatever they are, do they spread their religion WITH their knowledge of maths and sciences, or do they accept everyone, turning Mesopotamia into a multicultural/multi-religious hub?
If they spread their Christianity, then maths and sciences probably won't be received in Europe with the same skepticism. If maths and sciences come to Europe through some kind of heathens (Mandaeans, Manichaeans, Buddhists, etc.), then they just might, and this way, we might see a similar situation to the one we did in the Medieval Period OTL.
It all depends really on how the Arabs enter the rest of the Middle-East. One that IS for sure, is that the Arabs had a large population, and HAD to move. There is no stopping that, not without an earlier POD. But where and how they move? Do they conquer anything at all, or are they accepted into the lands of the next Persian dynasty and the Eastern Roman Empire peacefully? Probably not, but it could happen.