Pope Innocent III († 1216) stated that a pope can wither away into heresy and not believe the Faith.

"The pope should not flatter himself about his power, nor should he rashly glory in his honour and high estate, because the less he is judged by man, the more he is judged by God. Still the less can the Roman Pontiff glory, because he can be judged by men, or rather, can be shown to be already judged, if for example he should wither away into heresy, because "he who does not believe is already judged." (St. John 3:18) In such a case it should be said of him: 'If salt should lose its savor, it is good for nothing but to be cast out and trampled under foot by men. '" (Sermo 4)

Pope Adrian VI († 1523) stated that it is beyond question that a pope can err in matters touching the Faith, he can teach heresy in decrees. He also stated many Roman Pontiffs were heretics.

"If by the Roman Church you mean its head or pontiff, it is beyond question that he can err even in matters touching the faith. He does this when he teaches heresy by his own judgment or decree. In truth, many Roman pontiffs were heretics. The last of them was Pope John XXII († 1334)."

(Quaest. in IV Sent.; from Papal Infallibility and the Syllabus, 1908 by Viollet).*

(* According to the 1907 Catholic Encyclopedia, this work was published in 1512 from the notes of his student and without his supervision, but as it saw "many editions" it would appear that the pope did not repudiate the passage as not his own, in a work attributed to him.)

Venerable Pope Pius IX († 1878) recognized the danger that a future pope would be a heretic and teach contrary to the Catholic Faith, and he instructed, do not follow him.

"If a future pope teaches anything contrary to the Catholic Faith, do not follow him." (Letter to Bishop Brizen)

Pope Adrian II († 872) admitted that papal heresy renders lawful the resistance of subordinates to their superiors, and their rejection of the latter's pernicious teachings.

"We read that the Roman Pontiff has always possessed authority to pass judgment on the heads of all the Churches ( i.e., the patriarchs and bishops ), but nowhere do we read that he has been the subject of judgment by others. It is true that Honorius was posthumously anathematized by the Eastern churches, but it must be borne in mind that he had been accused of heresy, the only offense which renders lawful the resistance of subordinates to their superiors, and their rejection of the latter's pernicious teachings".

However, I must disagree with Pope Adrian when he said that heresy was the only offense that justified resistance: the Saints and Doctors have informed us otherwise, as we shall see.

Further, Pope Honorius I († 638) was not merely accused of heresy or anathematized by the Eastern Churches: he was anathematized as a heretic by the ecumenical Council of III Constantinople, whose Acts were confirmed by Pope Leo II († 683).

"We foresaw that, together with them, also Honorius, before Pope of Old Rome, is cast out of the Holy Catholic Church of God and anathematized, for we have found by his writings sent to [the heretic] Sergius, that he followed the thinking of the latter in everything, and continued his impious principles. [...] To Sergius, the heretic, anathema! To Cyrus, the heretic, anathema! To Honorius, the heretic, anathema!"

So we see that popes have told us that a pope can wither away into heresy and not believe the Faith; that it is beyond question that a pope can err in matters touching the Faith, he can teach heresy in decrees; that many Roman Pontiffs were heretics; that a pope may be a heretic and teach […] contrary to the Catholic Faith, in which case we are to follow the instruction do not follow him; and that papal heresy renders lawful the resistance of subordinates to their superiors, and their rejection of the latter's pernicious teachings.

The first Doctor of the Church, St. Athanasius († 373), told us that Catholics faithful to Tradition can be reduced to a handful. He wrote during the Arian crisis, when the global episcopacy defected to Arianism and Pope Liberius († 366) went into heresy, signed a heretical Arian creed and invalidly excommunicated St. Athanasius, as did the heretical bishops of the East.

"Even if Catholics faithful to Tradition are reduced to a handful , they are the ones who are the true Church of Jesus Christ." (Epistle to the Catholics)

St. Vincent of Lerins († 445) is the Father of the Church most associated with the defense of unchanging doctrinal tradition. It is the subject of his main treatise, the Commonitory. He foresaw that if the whole Church should go into heresy we must keep to the traditional Faith handed down from the Fathers.

"What then should a Catholic do if some portion of the Church detaches itself from communion of the universal Faith? What choice can he make if some new contagion attempts to poison, no longer a small part of the Church, but the whole Church at once? Then his great concern will be to attach himself to antiquity which can no longer be led astray by any lying novelty." (Commonitory)