This is quite a tricky question, and I don't have any sources on hand right now, but generally, we in the west have a dichotomy of the Muslim world being a distinct other with a completely different tradition to our own. This isn't necessarily true. The Muslim world's classical philosophy and writers are the same as ours. Socrates and Plato were as much the progenitor of their civilization as they are of ours. As such, the Islamic world generally had a favorable view of the antique Roman Empire, because it was seen as a predecessor civilization to their own.

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Now moving on to the Empire post-Heraclius, the Islamic world was in awe of it. Islamic visitors to Constantinople were just as awed and enamored of it as were the western visitors. The Islamic world viewed the Byzantine Empire as sort of the metric by which they measured their own "progress". An early hadith reported that when Muhammad was asked which city, Rome or Constantinople, would be conquered first, he replied "the city of Heraclius". Another early hadith references "the Qaisar who beat the Magus Kasra (Khosrow?) who was the King of Persia". Another reports the prophet as having said, "verily shall ye conquer Constantinople, and blessed will be the emir [of Constantinople] and blessed be it's army". The fortunes of Byzantium to some extent fluctuated with those of the various Islamic states of the mediterranean.