Hi, I specialize in the Byzantines but I have read quite a bit on the Arab conquest of the Middle East.

The transition was at first, a social revolution, not so much of a religious one.

In fact, when the Muslims came, the Holy Land and the Middle East contained a plethora of Christian denominations, with the state Orthodox christianity of the Roman/Byzantine Empire being the largest. As the population in the Holy land was predominantly non-Greek, it often viewed Orthodox Christianity associated with the ruling class of Greeks, who made up the core of the empire's population at that time. The empire itself was better off when unified, religion-wise, thus all sects and heresies of Orthodox Christianity were often persecuted, with government approval. Many times there were schisms and even armed revolts due to the differences between the many sects and the state Orthodox church. Other than various types of Christianity, Judaism was also tolerated, but the lives and rights of the jews were highly regulated.

Despite the Western Empire falling, the Eastern Roman Empire managed to retain the richest parts of the empire, thus the Imperial administration continued to function without interruption. This was a very effective administration for its time, capable of meticulously levying taxes from the population. These taxes, however, were pre-determined and non-flexible. Thus a natural disaster or war devastation for example, could lead to the affected people being forced to pay the pre-determined amount of tax, leaving them often with nothing to provide themselves with, due to the aforementioned special circumstances. Many subject peoples viewed these taxes as another form of Imperial oppression and this was most likely the case with the native population of the Holy Land as well. Especially in Egypt, the Copts saw Greeks as the overlords and the Orthodox faith as a form of persecution against their Coptic church.

The Byzantine-Sasanian war of 602-628 changed many things to the worse. The already substantial tax burden levied by the Imperial government was enlarged even further. Armies of both sides ravaged through Middle East and the Holy Land. The war left the empire weakened and the Holy Land ripe for the taking.

As the Muslims conquered the Levant in the first half of the 7th century, most of the population remained Christian. Their conquest of the Byzantine Empire stopped at the gates of Anatolia, much due to the reasons I state in this answer.

The muslims didn't seek to actively convert the population. Instead, they forced the people of the book, Christians and Jews, to pay the poll tax called Jizya, which they used to finance their state administration and armies. For the population of the area, this was actually a relief, for the Jizya was much lower than the Imperial tax levied by the overlords of Constantinople. The situation for the various of Christian sects also improved. No longer did the Orthodox church rule over them all, but under the Muslim Caliph, all of the different Christian denominations were treated equally. This went for the Jews as well, whose situation was improved significantly by the Muslim conquest. This kind of tolerance was a vital instrument in keeping the Caliphate together. It contributed greatly towards the loyalty of these subjects to their Muslim rulers. There was some resistance towards Muslim rule, especially regarding the many major churches which were converted into mosques. However, this resistance was never major enough to challenge Muslim rule and the major Christian threats came from outside, mostly from the direction of the Byzantines. In fact, Muslim rulers of the Holy Land and the Middle East often used Christian soldiers in their armies and especially in their navies.

Needless to say, although the Jizya was a moderate tax, it was a tax still. In time, it prompted the population to slowly convert. This conversion was gradual and began from those who wished to obtain a high status in their society.

As for influences the Byzantine Empire had on the Islamic States, certainly, administrative structures were passed on. The tax collection system was inherited from the Byzantine authorities and this was used to collect the Jizya. The muslims were especially keen to imitate Byzantine successes in warfare and tactics. Emperor Leo VI makes a few remarks of this in his military manual Taktika (early 10th century), where he states that the Saracen heavy cavalry is almost as good as Roman, but not quite, having adapted Roman tactics and equipment to their own use.

As a conclusion, we can say that the Muslim conquerers imposed a policy of tolerance in the Holy Land on a population which had been subjected to heavy Imperial tax and sometimes ethno-religious persecution under Imperial Roman/Byzantine rule. This led to an easy transition of power in the Middle East, in the conqueror's perspective, and paved the way for a gradual conversion of these lands to Islam.

I hope this answered your question. For reading, I recommend The Grand Strategy of the Byzantine Empire by Edward Luttwak. It quite thoroughly displays what went wrong in the Byzantine administration of the Middle East and I've used it as a source many times. Warfare, State And Society In The Byzantine World 565-1204 by John Haldon might also offer some good insights on Byzantine-Arab relations. For contemporary sources, see The Taktika of Leo VI, attributed to the Byzantine Emperor Leo VI and translated in english by George T. Dennis.