Pope Honorius' heresy deprived him of the papacy, if you follow the teachings of the fathers and innumerable theologians.
A heretical pope ceases not only to be pope but being a member of the Church.
St. Robert Bellarmine (De Romano Pontifice, II.29.):
Just as it is licit to resist the Pontiff who attacks the body, so also is it licit to resist him who attacks souls or destroys the civil order or above all, tries to destroy the Church. I say that it is licit to resist him by not doing what he orders and by impeding the execution of his will. It is not licit, however, to judge him, to punish him, or to depose him, for these are acts proper to a superior.
Do you believe in obeying error?
Bellarmine and Baronius followed Pighius in denying that Honorius was condemned at all.
The passage cited is from a lengthy chapter Bellarmine devotes to refuting nine arguments advocating the position that the pope is subject to secular power (emperor, king, etc.) and an ecumenical council (the heresy of conciliarism).
The general context, therefore, is a discussion of the power of the state vis-à-vis the pope...In its particular context, the oft-cited quote is part of Bellarmine's refutation of the following argument:
Argument 7. Any person is permitted to kill the pope if he is unjustly attacked by him. Therefore, even more so is it permitted for kings or a council to depose the pope if he disturbs the state, or if he tries to kill souls by his bad example.
Bellarmine answers:
I respond by denying the second part of the argument. For to resist an attacker and defend one's self, no authority is needed, nor is it necessary that he who is attacked be the judge and superior of him who attacks. Authority is required, however, to judge and punish.
It is only then that Bellarmine states:
Just as it is licit to resist the Pontiff who attacks the body, so also is it licit to resist him who attacks souls or destroys the civil order or above all, tries to destroy the Church. I say that it is licit to resist him by not doing what he orders and by impeding the execution of his will. It is not licit, however, to judge him, to punish him, or to depose him, for these are acts proper to a superior. (De Romano Pontifice, II.29.)
Bellarmine...is discussing the course of action which may legitimately be taken against a pope who upsets the political order or "kills souls by his bad example." A king or a council may not depose such a pope, Bellarmine argues, because they are not his superior—but they may resist him.
Nor does this quote support those traditional Catholics who would recognize John Paul II as pope but reject his Mass and ignore his laws.
First, the passage justifies resistance by kings and councils. It does not say that individual bishops, priests and laymen on their own possess this right to resist the pope and ignore his commands—still less that they can set up places of worship in opposition to diocesan bishops a pope has lawfully appointed.
Second, note the precise causes for resistance in the case Bellarmine is discussing: disturbing the state or giving bad example. These, obviously, are not the same thing as papal liturgical legislation, disciplinary laws or doctrinal pronouncements which an individual might somehow deem harmful. Bellarmine would hardly approve of disregarding, carte blanche, for 30 years the directives of men one claims to recognize as legitimate occupants of the papal office and the vicars of Christ on earth.
In sum, the passage neither condemns sedevacantism nor supports traditionalists like the adherents of the Society of St. Pius X.
*"trads," in misusing Bellarmine's quote, taken out of context, advance and solidify, in the minds of those disobedient to the Vicar of Christ, the idea that opposition is not only permitted but that such opposition is principled