The short answer is that it kind of depends.

Scholars for a while have been talking about how non-Christians were often more than happy to serve alongside Christians (and vice-versa) at the time of the Crusades -- see, for example, Prof. Hussein Fancy's recent work. This happened not only in Iberia but also in the Near East as well. One of the major issues with the 2nd Crusade, for instance, was that the crusaders decided to attack Damascus, which at the time was allied with the Kingdom of Jerusalem and aiding them against Zengi (among others). That doesn't really answer your question though.

Certainly, native Eastern Christians aided the crusaders throughout their time in the Near East. The Armenians and citizens of Edessa were instrumental in helping the First Crusade proceed through Anatolia and into Palestine. The aid that Baldwin was able to send, via the Edessans, to the crusade outside Antioch probably saved that expedition. There's also a great bit from the chronicler Fulcher of Chartres that talks about the intermingling of cultures once the Latins were established in the East -- how they began to "go native." That, I think, speaks to a sense of cooperation among the various Christian groups they encountered.

Later on, Byzantines would at times fight alongside crusaders on various expeditions and local Syriac Christians seem to have intially been quite grateful for the arrival of their Western co-religionists. That began to change later, however, and there seems to be some evidence that they were relatively indifferent once the so-called "Crusader States" began to melt away in the 13th century. The Copts don't seem to have been particularly helpful to the crusaders during the various Christian expeditions into Egypt and may have actively aided the Fatimids against the Kingdom of Jerusalem, thereby demonstrating that they thought themselves to be more "Egyptian" than "Christian" (if that makes sense).

If you'd like to know more, the best book on this is by far Christopher Macevitt's Rough Tolerance.