I see much speculation in this thread. Instead, we should be looking at what the canons of the Church say. More specifically, the initial poster asks the question:
If it ever becomes apparent that a Patriarch is an open and manifest heretic, like a gnostic or espouses Nestorian heresies, is it my responsibility as a layman under his eparchial synod to sever communion with him?Fifteenth Canon of the First-Second Synod
If, however, a Bishop, Metropolitan, or Patriarch begins to preach publicly in Church any heretical doctrine that is antithetical to Orthodoxy, then the aforementioned clergy have a right and at the same time an obligation to separate themselves forthwith from that Bishop, Metropolitan, or Patriarch, and for this reason not only will they not be subject to any canonical penalty, but will, moreover, even be praised, insofar as they have not thereby reprehended or rebelled against legitimate Bishops, but against false bishops and false teachers, and have not thereby initiated any schism in the Church, but on the contrary have delivered the Church, as far as possible, from schism and division.
Archimandrite John (a well-known Russian Canonist [later Bishop of Smolensk—Trans.]), having in view the historical circumstances of the Church of Russia, observes, in his interpretation of the present Canon, completely correctly and in accordance with a rigorous conception of Canonical science, that a Presbyter who secedes from his own Bishop by reason of heretical teaching [on the part of the latter] will not be culpable, but praiseworthy, yet when and only when the said Bishop begins preaching a doctrine that openly contradicts the teaching of the Orthodox Church and has been formally condemned by the Church (6) and if he proclaims this false doctrine openly, publicly, and in Church, with resolute intent to subvert Orthodox teaching and to uphold heresy; if otherwise (if, that is, a Bishop expresses some private opinion of his on matters of faith and morals which might appear to someone as incorrect, if the Bishop in question expresses his erroneous doctrine in a narrower circle of particular persons, so that it is attainable that the doctrine in question be corrected within this narrower circle, the peace of the Church not being breached), in such a case no Presbyter has the right to secede high-handedly from his own Bishop and create a schism; otherwise, he will be subject to the injunction prescribed regarding these matters by the Thirty-first Apostolic Canon.
(Related Canons: Thirty-first Apostolic Canon; Sixth Canon of the Second OEcumenical Synod; Third Canon of the Third OEcumenical Synod; Eighteenth Canon of the Fourth OEcumenical Synod; Thirty-first and Thirty-fourth Canons of the Synod in Trullo; Sixth Canon of the Synod of Gangra; Fourteenth Canon of the Synod of Sardica; Fifth Canon of the Synod of Antioch; Tenth, Eleventh, and Sixty-second Canons of the Synod of Carthage; Thirteenth and Fourteenth Canons of the First-Second Synod).
• “Note that the present Apostolic Canon decrees that clergy may without peril secede from their Bishops if they reprehend them for wrong belief” (Theodore Balsamon, Patrologia Græca, Vol. CXXXVII, col. 97C [commentary on the Thirty-first Apostolic Canon]).
• The faithful of Constantinople, both clergy and people, walled themselves off from Patriarch Nestorios prior to any Synodal judgment, because this man was preaching a newfangled and reprehensible heresy; that is, “they reprehended a false bishop and false teacher.”Source: http://www.synodinresistance.org/pdfs/2010/12/E3a3b004Ermhneia15-1&2%20Folder/E3a3b004Ermhneia15-1&2.pdf
- Added by PtA after receipt from poster