David Larison challenges this interpetation:
...it is my firm opinion that the myth of collaboration, specifically monophysite collaboration, with the Islamic invader is unfounded in the evidence and anachronistic in its interpretation of the attitudes of the Roman Christians whose loyalty to their ecumenical polity is being traduced. Very simply, there is no evidence of monophysites collaborating with the Islamic invader.
I would like to know if the popular interpretation itself derives from Edward Gibbon.
Daniel Larison responds to my Gibbon query:
As far as I know, Gibbon was the most well-known early advocate of the view that the heretics opened the door to the Muslims. For him, this may have made more sense in his age when fairly recent British history (and to some extent even in his own time) had seen religious loyalties dictating political allegiances in a very dramatic and decided way. What Gibbon could not, or did not want to see (or perhaps literally could not without access to some of the Coptic and other Oriental sources) was that serious Christians still considered themselves to be Romans and believed themselves to be foremost supporters of the empire who would do nothing consciously to undermine it. Since a large part of his explanation of the decline focuses on the role of Christianity and the change this wrought (not entirely wrong, but far too oversimplified and polemical in the way he explains it), he did not see continuity in the attitudes of the Roman people themselves or considered such attitudes less important than the "weaknesses" their Christianity gave the empire. It is odd, but this is one of the oldest Gibbonian holdouts in later Roman history that Byzantinists have not repudiated, and it is one of the few that many in our field endorse. But that is changing, and that is the good news.