People often confuse these two points when considering the current situation as it relates to sedevacante. Many seem to believe that the only way to know if a pope has lost office is if he teaches a heresy from what would otherwise be his infallible capacity, that is if he tries to teach heresy ex cathedra. This is not at all the case. All that would be necessary for the pope to lose office is for him to publicly express a heretical opinion.
Here is where anti-sedvacantists are up in arms, however. They point to certain popes in the past who have either taught erroneously, or have not come down as hard as they should on heresy. But the fact is that an erroneous doctrine, if it is not contrary to a teaching that is to be believed by Divine and Catholic Faith (dogma) is not heresy, and one holding such an opinion, though they may be sinning by doing so depending on how certain the belief is regarded to be and how much support it has. Even if his belief is condemned as heretical later on in history, he does not lose his membership in the Church for holding it prior to that time, nor does he lose any authority he holds within Her. Heresy was always false, but not necessarily always heresy.
Now let's delve into the specific cases that are so often presented (or rather misrepresented):
Overcoming objections concerning Pope Liberius:
Pope Liberius is said by opponents to papal infallibility to have signed a semi-Arian creed, one that rejects the Arian term “homoiousios”, which means of like substance, while also rejecting the correct Catholic term “Homoousion”, which means of the same substance (that the Son of God is of the same substance as the Father), thereby falling into heresy.
It’s said that Athanasius confirms this in his writings, however, examining his writings shows this is actually not the case.
St. Athanasius, History of the Arians, Part V, #35: “The eunuch accordingly went to Rome, and first proposed to Liberius to subscribe against Athanasius…”
The Arians wanted Pope Liberius to condemn St. Athanasius, they did indeed also wanted the pope to condemn the "Homoousion".
St. Athanasius, History of the Arians, Part V, #41: “But Liberius after he had been in banishment two years gave way, and from fear of threatened death subscribed. Yet even this only shows their violent conduct, and the hatred of Liberius against the heresy, and his support of Athanasius, so long as he was suffered to exercise a free choice. For that which men are forced by torture to do contrary to their first judgment, ought not to be considered the willing deed of those who are in fear, but rather of their tormentors. They however attempted everything in support of their heresy, while the people in every Church, preserving the faith which they had learned, waited for the return of their teachers, and condemned the Antichristian heresy, and all avoid it, as they would a serpent.”
St. Athanasius relates not that Pope Liberius signed a heretical creed, but that he signed a condemnation against Athanasius, and that under extreme duress. While this is indeed bad, it is not a profession of heresy, and St. Athanasius would certainly not leave a detail such as the pope falling into Arian or semi-Arian heresy out of his "History of the Arians".
Overcoming objections concerning Pope Vigilius:
The objection is occasionally made that Pope Vigilius had erred regarding his condemnation of the 'Three Chapter' and his subsequent withdrawal and re-condemnation of the same.
According to A History of Christendom, vol.2 The Building of Christendom by Warren H. Carrol pages 173-178:
544 - Emperor Justinian issues a decree condemning the Three Chapters and the four Eastern patriarchs, lead by Mennas of Constantinople, give assent to the decree on the condition of the Pope's approval.
November 2, 545 - Pope Vigilius is arrested for not approving the decree
Fall of 546 - The Pope is brought to Greece when it becomes clear he will not approve the decree
January 547 - The Pope arrives in Constantinople. Lifts his earlier excommunication of Mennas for even conditionally assenting to Justinian's decree and gives private assurances to Justinian he will issue a document.
April 548 - Issues "Judicatum" condemning the person and writings of Theodore of Mopsuestia, and the heresies in the writings of Theodoret of Cyrrhus and Ibas, without prejudice to Chalcedon. In reaction to this, he is excommunicated by a synod of African bishops (who in reality would have excommunicated themselves, since the pope cannot be excommunicated, save if he excommunicates himself through heresy).
August 550 - Vigilius convinces Justinian to allow him to withdraw Judicatum and have a council to decide the matter. In exchange, Justinian demands Vigilius' promise in writing that he will do what he can to convince the council to condemn the Three Chapters.
July 551 - Western bishops state they will not attend the proposed council or if they did attend they would not vote for condemnation. Justinian issues a second statement condemning the Three Chapters and insists the Eastern bishops sign it. Vigilius announces excommunication for any bishop that signs the statement. Many bishops sign and are excommunicated. This is done because it is the Pope and the council that decide issues of orthodoxy and not the Emperor. In reaction the Emperor places Vigilius under closer guard at the Placidia Palace.
December 23, 551 - Vigilius escapes and takes refuge in Chalcedon.
February 5, 552 - Vigilius issues an encyclical denouncing his treatment by Justinian and professing his faith
March 552 - Justinian sends Mennas and Theodore Askidas to Chalcedon to apologize for his mistreatment and pledging fidelity to the Council of Chalcedon and support for the Pope in suppressing the controversy until the council met. Vigilius returns immediately to Constantinople but is not allowed to return to Rome.
May 5, 553 - The Council of Constantinople convenes. Only 166 bishops attend. Because he was unable to hold a preliminary synod in Rome and the small representation of Western bishops, Vigilius refuses to attend but issues a "Constitutum" reaffirming Chalcedon, condemning the works but not the person of Theodore of Mopsuestia, upheld Theodoret of Cyrrhus and Ibas of Edessa, condemned five heretical Nestorian propositions, and directed further discussion of the matter to cease. In response, Justinian issues Vigilius' written promise to attempt to convince the council to condemn the Three Chapters'. The council takes up the matter.
June 2, 553 - The council anathematizes the Three Chapters but upholds Chalcedon. Theodore of Mopsuestia's works and person were condemned, only Theodoret of Cyrrhus' heretical writings are condemned as well as the letter of Ibas of Edessa. They did not decide the issue of whether Ibas of Edessa actually wrote the letter which he apparently denied. Vigilius' name was struck off the diptychs.
June to December, 553 - Vigilius along with his staff are put in close confinement on a diet of bread and water on the island of Proconnesos. During this Vigilius suffered from kidney stones from which he later died on the way back to Rome.
February 554 - Vigilius issues a final Judicatum ratifying the Council of Constantinople, because he said after study and reflection he had come to see how much heresy, whether or not intended, was indeed included in the writings.
None of the heresies were specifically approved by the Council of Chalcedon. The authority of the pope had been upheld because even the action of the Emperor and a council could not decide the issue without the ratification of the pope. No fundamental dogma of the Faith was involved in the Three Chapters controversy. Therefore, Vigilius ratified the council.
June 7, 555 - Pope Vigilius dies in agony from kidney stones in Syracuse, Sicily on his way back to Rome after being released from ten years of exile by Emperor Justinian.
Nowhere did the councils or the pope teach error or heresy. Therefore, there is no reason to call the dogma of papal infallibility into question. After all, the dogma of papal infallibility does not state the pope will never make mistakes, only that he will be prevented from teaching any error or heresy at all when speaking ex cathedra.
Overcoming objections concerning Pope Honorius:
Pope Honorius, always the last resort for those attempting to attack papal infallibility. Pope Honorius lived at the time the Monothelite heresy was gaining adherents. As Pope it was his duty to speak out forcefully against this false doctrine. He failed to do so. In addition, he wrote a poorly worded, rather ambiguous letter to the Patriarch of Constantinople, attempting to respond to an equally ambiguous letter received from the Patriarch concerning the heresy, which the Patriarch endorsed.
The ambiguity and lack of forcefulness of Honorius' letter was unfortunately used by the Monothelites to strengthen their movement by suggesting that the Pope was a supporter. Honorius was not very careful in accurately and fully expressing the position of the Church, and not very straightforward in his condemnation of the heresy.
However, none of the above could categorize him personally as a heretic - a categorical impossibility since he was the Vicar of Christ. Failure to speak out forcefully in condemnation of a heresy does not make one a heretic. Personally holding a view that is objectively heretical does not make a prelate a heretic until his view is externally and publicly expressed, and anyway there is no evidence that Pope Honorius held such a view.
And nothing expressed in a personal letter, whether misinterpreted by enemies of the Church or even accurately interpreted, could make one a heretic. Pope Honorius would have been a heretic if he had officially endorsed Monothelitism. He didn't even come close. It is true that after his death he was among those declared anathema by the Council of Constantinople the Third in its condemnation of Monothelitism; however, the decrees of that council, like any council, are not valid until confirmed by the Pope. Before Pope Leo II confirmed the findings of the Council, he modified the references to Pope Honorius, specifically to make it clear that Honorius had not supported the heresy of Monothelitism, but was only negligent in his duty to actively suppress it. Negligence, yes. Heresy? Not even close.
Overcoming objections concerning Pope John XXII:
In the last years of Pope John's pontificate there arose a dogmatic conflict about the Beatific Vision, which was brought on by himself, and which his enemies made use of to discredit him.
Before his elevation to the Holy See, he had written a work on this question, in which he stated that the souls of the blessed departed do not see God until after the Last Judgment. After becoming pope, he advanced the same teaching in his sermons. In this he met with strong opposition, many theologians, who adhered to the usual opinion that the blessed departed did see God before the Resurrection of the Body and the Last Judgment, even calling his view heretical.
A great commotion was aroused in the University of Paris when the General of the Minorites and a Dominican tried to disseminate there the pope's view. Pope John wrote to King Philip IV on the matter (November, 1333), and emphasized the fact that, as long as the Holy See had not given a decision, the theologians enjoyed perfect freedom in this matter.
In December, 1333, the theologians at Paris, after a consultation on the question, decided in favour of the doctrine that the souls of the blessed departed saw God immediately after death or after their complete purification; at the same time they pointed out that the pope had given no decision on this question but only advanced his personal opinion, and now petitioned the pope to confirm their decision. John appointed a commission at Avignon to study the writings of the Fathers, and to discuss further the disputed question.
In a consistory held on 3 January, 1334, the pope explicitly declared that he had never meant to teach aught contrary to Holy Scripture or the rule of faith and in fact had not intended to give any decision whatever. Before his death he withdrew his former opinion, and declared his belief that souls separated from their bodies enjoyed in heaven the Beatific Vision.
In his papal bull Benedictus Deus, Pope John’s successor, Benedict XII defined the dogma that the souls of the faithful departed go to their eternal reward immediately after death or Purgatory.
If Pope John had publicly taught his erroneous opinion after the contrary had been defined by the Holy See, then he would have been a heretic who lost office, but this, again, is a simple case of a pope teaching (without the intention of binding the faithful) an erroneous opinion that was not at that time opposed to a formal dogma.
What Must You Do To Get to Heaven?