There wasn’t much impact on the “eastern” Christians who already lived there. There were various sects of Christians living in Muslim territory - Greek Orthodox, Syrian Orthodox, Armenians, Maronites in the mountains of Lebanon, Copts in Egypt, followers of the Church of the East in Iraq and Persia, and further away on the fringes of the Muslim world there were Georgians in the north and Nubians and Ethiopians in Africa. The Latin Catholic crusaders who invaded northern Mesopotamia, Syria, and Palestine mostly encountered Greeks, Syrians and Armenians.

The Greeks and Armenians in Anatolia and Mesopotamia helped the crusaders and sometimes turned towns and cities over to them - that’s essentially how the first crusader state was founded, in Edessa. Antioch was ruled by Muslims and the crusaders had to besiege it, but it fell because the Christian citizens secretly let the crusaders in.

To the south, the Seljuks of Syria and the Fatimids of Egypt were fighting amongst themselves and weren’t too concerned with the crusade. They probably assumed it was a regular Byzantine army at first, since the Byzantines often fought military campaigns in Syria, although both the Fatimids and Seljuks eventually realized that this time, Jerusalem was the target. The Fatimids took Jerusalem from the Seljuks in 1098, while the crusaders were on their way.

The Fatimid governor of the city knew that the Christians in the north had been helping the crusaders, and the ones around Jerusalem were doing so too (they invited the crusaders to occupy Bethlehem, for example). So to avoid any treachery by the eastern Christians he supposedly expelled them all from Jerusalem. The reports might be inaccurate though, or some of them remained in the city anyway, because once the crusaders took Jerusalem and started massacring the inhabitants they found that they couldn’t distinguish between the eastern Christians and Muslims and had accidentally killed a lot of Christians as well. Also, the Armenians apparently joined the crusaders in attacking the Muslims. So it’s kind of confusing what happened! Some of them may have been expelled, some of them were clearly still inside.

In the years after the First Crusade the Muslims of the Near East figured out that this was something different, not a simple Byzantine campaign, but a religious war for the holy sites in Jerusalem and possibly even a coordinated attack on Muslims all over the world, in Spain and North Africa too. (This was actually a somewhat accurate assessment, but the Muslims vastly overestimated the amount of coordination!)

Muslims didn’t care much about the differences between various sects of Christians, but clearly it was only the Latin-speaking Franks from across the sea who were the problem. The native eastern ones weren’t typically punished for things these new Latin Christians did. As long as they didn’t rise up against their Muslim rulers and they continued to pay their taxes, there was no reason to target them.

The only community that was really affected were the Copts in Egypt. They were typically loyal to the Muslim government, at least when the Fatimids were in charge. They served the government especially in financial positions, but they were also known to be excellent physicians. When the crusaders invaded Egypt in the 1160s, they had no help from the Copts like they did from the Greeks and Armenians during the First Crusade and they generally thought the Copts were untrustworthy. Unfortunately for the Copts, the Ayyubids of Syria didn’t think they were very trustworthy either. The Ayyubids (the dynasty of Saladin) conquered Egypt from the Fatimids and associated the Copts with the previous Fatimid government, so the Copts began to be marginalized a bit - they couldn’t hold administrative positions anymore, they couldn’t display crosses or ring church bells, or build new churches for example. But this probably has little to do with the crusades directly (well, aside from the fact that the Ayyubids were in Egypt only because the crusaders had invaded it). In the other Ayyubid territories in Syria the Christians don’t seem to have been affected in this way.

So, there really wasn’t much impact. Some Christians may have been expelled from Jerusalem just before the crusader siege, but otherwise they were distinguished from those strange violent barbarian Franks from across the sea, and they weren’t punished for things the crusaders did.

Sources:

Paul M. Cobb, The Race for Paradise: an Islamic History of the Crusades (Oxford University Press, 2014)

Thomas Asbridge, The First Crusade: A New History (Oxford University Press, 2004)

Yaacov Lev, Saladin in Egypt (Brill, 1999)