Full quote below from Empires and Barbarians pages 380-381. The point is by no means that it became an irrelevance, but stopped being a superpower and its success became very dependent on what was happening with the more powerful Islamic world it was adjoined to.
"From the early seventh century, Constantinople was no longer a pan-Mediterranean power and major player on the broader European stage. Though still important in the eastern Mediterranean it became in many ways an unwilling satellite state of the Islamic world, no longer substantially in charge of its own fate. Its subsequent periods of prosperity and decline correlate closely and inversely with the history of the new Islamic power block. When Islam was politically united, Constantinople was condemned to decline; when - as sometimes happened - Islam itself fragmented, there was room for modest expansion. In short, the self-proclaimed imperial Romanness of the rulers of post-seventh-century Constantinople is a chimera. The losses suffered at the hands of Islam meant that these emperors were now ruling what was as much a successor state to the Roman Empire as any of the new powers of the Roman West a century earlier. My own preference, in fact, is to use the term 'Byzantine' rather than 'east Roman' from the mid-seventh-century, as a reflection of how great a sea change the rising tide of Islam had created in Mediterranean history'
I've kept this page open and not answered in hopes of seeing someone with more knowledge than I step in, but I see not much of a response so far. Personally, I can't say that I agree with Heather's take. While their relations with their Muslim neighbors were obviously essential to the history of the ERE, what Heather is essentially contending is that for an 800 year period which saw declines, resurgences, dynamism and massive evolution in policy and leadership can be reduced to the effects of Muslim expansion and contraction. This, to me, feels very much out of the old school of Western scholarship on Byzantium which treated it as essentially a state in constant decline.
One interesting consideration whenever we talk about Byzantine-Muslim relations is that the Muslim perspective is often more or less absent from Western scholarship. Part of this can be attributed to the limited accessibility of medieval Arabic sources to scholars with knowledge of mainly Western languages, but in part I think it's a result of the cultural overtones present in Western history scholarship. Byzantium was a powerful figure in the eyes of Muslims--it held a nearly-mystical aura and was indeed idealized by many of them in the same way that it amazed medieval Christians seeing Constantinople for the first time in the Crusades. As far as I know there is only one really good English-language work synthesizing the Arab view of Byzantium. That is Nadia Maria El-Cheikh's Byzantium Viewed by the Arabs. In it, she argues that, "Byzantium was also an imaginary category of difference against which Islam defined itself" (232). This certainly doesn't sound like the perspective that a dominant imperial power would have of a satellite state.
Unfortunately, I'm presently a continent away from my books on the ERE so I won't be able to pull exact quotes as easily for the rest of this answer. However, I will do my best to respond more specifically to Heather's claims from the Byzantine point of view as well. Some of the most defining events of the middle years of the ERE had nothing whatsoever to do with the spread of Islam. Pre-Manzikert, Constantinople conceived of themselves as powerful enough to fight back against the Muslims independently. Borders were pushed back and forth across Anatolia for centuries. A number of historians contend that Manzikert represented the real turning point in decline--that the loss of its Anatolian heartland robbed the empire of critical supplies and led to military weakening.
To get back to what I see as the reductionist elements of Heather's argument, the 7th century was a profound period of change in Byzantium regardless of the rise of Islam. It really represented a broader transition in Eastern Roman society--the final ascendancy of Orthodoxy as the dominant religious expression throughout the Empire, a complete reorganization of the military structure away from classical Roman lines, the demise of classical Hellenic urban culture, etc. That the first period of Iconoclasm arose in the very early 8th century is no coincidence--it was a reaction to the massive change engulfing the empire as a whole, most of which was happening entirely without regard to the changing political situation in the Levant. This is likely where Heather's notion that the Roman-ness of the Byzantines was chimerical after this period, but I again think that's a bit of a simplification. There were always significant differences culturally between the Hellenic eastern empire and the more traditional western empire. Haldon provides a pretty comprehensive analysis of this transformation in Byzantium in the 7th Century: Transformation of a Culture, which I highly recommend.
In Byzantium and the Early Islamic Conquests, Walter Kaegi presents quite clearly the idea that in fact the rapid spread of Islam in the 7th century was in part the result of a tactical military decision on the Byzantines' part. While the Levantine provinces were quite valuable, Syria was incredibly difficult to hold militarily in the face of mounting Islamic pressure and so the empire pulled back to focus its resources on defending its core regions that were more vital to its interests, from which it continued to prosper. In a sense, he argues that the loss of Syria was more psychologically than tactically damaging.
I realize this answer is getting both long and a bit divergent, so I'll wrap up by leaving myself open to any specific questions that you might have. But in summary/tl;dr: Heather's claim is overly reductionist and fails to account for the much larger scale of change experienced across the Eastern Roman Empire during the 7th century, as well as its subsequent dynamism (although expressed across a smaller geographic extent than at its peak under Justinian).