Heresy

Heresy consists in the formal denial or doubt by a baptized person of any revealed truth of the Catholic Faith; as a crime it consists of the outward and pertinacious manifestation of the sin.

As we can see below, Scripture speaks frequently of heresy entering the Church. We can also see that heresy is possible even with a Pope. In that case, the Pope is immediately ejected from the Church and papal infallibility no longer applies. See quotes below.

References to Heresy in Scripture:

Take heed to yourselves, and to the whole flock, wherein the Holy Ghost hath placed you bishops, to rule the church of God, which he hath purchased with his own blood. I know that, after my departure, ravening wolves will enter in among you, not sparing the flock. And of your own selves shall arise men speaking perverse things, to draw away disciples after them. Acts 20:28-30

A man that is a heretic, after the first and second admonition, avoid: Knowing that he, that is such an one, is subverted, and sinneth, being condemned by his own judgment. Titus 3:10-11

Whosoever revolteth, and continueth not in the doctrine of Christ, hath not God. He that continueth in the doctrine, the same hath both the Father and the Son. If any man come to you, and bring not this doctrine, receive him not into the house nor say to him, God speed you. For he that saith unto him, God speed you, communicateth with his wicked works. 2 John 1:9-11

Now the Spirit manifestly saith, that in the last times some shall depart from the faith, giving heed to spirits of error, and doctrines of devils, Speaking lies in hypocrisy, and having their conscience seared...." 1 Tim 4:1-2

For there shall be a time, when they will not endure sound doctrine; but, according to their own desires, they will heap to themselves teachers, having itching ears: And will indeed turn away their hearing from the truth, but will be turned unto fables. 2 Tim 4:3-4

But there were also false prophets among the people, even as there shall be among you lying teachers, who shall bring in sects of perdition, and deny the Lord who bought them: bringing upon themselves swift destruction. And many shall follow their riotousnesses, through whom the way of truth shall be evil spoken of. And through covetousness shall they with feigned words make merchandise of you. Whose judgment now of a long time lingereth not, and their perdition slumbereth not. 2 Peter 2:1-3

Catholic Church Teaching on Heresy in the Church:

From the encyclical on "Unity of the Church", by Pope Leo XIII in 1896, "Our Lord Jesus Christ, when in His Gospel He testifies that those who not are with Him are His enemies, does not designate any special form of heresy, but declares that all heretics who are not with Him and do not gather with Him, scatter His flock and are His adversaries: He that is not with Me is against Me, and he that gathereth not with Me scattereth" (S. Cyprianus, Ep. lxix., ad Magnum, n. I)

Canon 33. "No one shall join in prayers with heretics or schismatics." The Synod of Laodicea (4th Century)

"Next comes the heretic who belongs to an heretical association; for such a person his heretical membership alone is sufficient to bring him under sentence of excommunication. In his case the penalty is incurred by adhesion to the heresy, notably by wilful and active participation in sacris (i.e. in public worship) with heretics" Catholic Encyclopedia: Excommunication (1917)

"There can be nothing more dangerous than those heretics who admit nearly the whole cycle of doctrine, and yet by one word, as with a drop of poison, infect the real and simple faith taught by our Lord and handed down by Apostolic tradition" Encyclical on Unity of the Church by Pope Leo XIII, 1896

Catholic Church Teaching on Heresy and the Pope:

In addition, [by this Our Constitution, which is to remain valid in perpetuity We enact, determine, decree and define] that if ever at any time it shall appear that any Bishop, even if he be acting as an Archbishop, Patriarch or Primate; or any Cardinal of the aforesaid Roman Church, or, as has already been mentioned, any legate, or even the Roman Pontiff, prior to his promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic Faith or fallen into some heresy:
(i) the promotion or elevation, even if it shall have been uncontested and by the unanimous assent of all the Cardinals, shall be null, void and worthless... "Cum ex Apostolatus Officio" Apostolic Constitution of Pope Paul IV, 1559

It shall be lawful for each and all of the cardinals,...as well as for all the clergy and the Roman people,... to withdraw without penalty and at any time from obedience and loyalty to the person so elected even if he has been enthroned (while they themselves, notwithstanding this, remain fully committed to the faith of the Roman church and to obedience towards a future Roman pontiff entering office in accordance with the canons) and to avoid him as a magician, a heathen, a publican and a heresiarch."
Pope Julius II, Council of Lateran V. 1513

"Now when [the Pope] is explicitly a heretic, he falls ipso facto from his dignity and out of the Church, and the Church must either deprive him, or, as some say, declare him deprived, of his Apostolic See."
St. Francis de Sales, "The Catholic Controversy"

"...a pope who is a manifest heretic by that fact ceases to be pope and head, just as he by that fact ceases to be a Christian and a member of the body of the Church; wherefore he can be judged and punished by the Church. This is the judgment of all the early fathers, who teach that manifest heretics immediately lose all jurisdiction."
St. Robert Bellarmine, "On the Roman Pontiff"

"If God permitted a pope to be notoriously heretical and contumacious, he would then cease to be pope, and the Apostolic Chair would be vacant."
St. Alphonsus de Liguori, "The Truths of the Faith"

"An heretical pope necessarily ceases to be head of the Church, for by his heresy he is no longer a member thereof: in the event of his still claiming the Roman see a general council, improperly so-called because without the pope, could remove him. But this is not deposition, since by his own act he is no longer pope."
A Catholic Dictionary, 1951. Deposition

"The councils of Constance and Basle, and Gallican theologians, hold that a council may depose a pope...(2) /ob fidem/ (on account of his faith or rather want of faith, i.e. heresy). In point of fact however, heresy is the only legitimate ground. For a heretical pope has ceased to be a member of the Church, and cannot, therefore, be its head."
Catholic Encyclopedia, 1913. [Vol. IV p.435] Councils

Summary

It is clear from verses in Scripture that the Apostles were swift and severe when it came to heresy in the Church. We can also see the same severity from the teaching of the Catholic Church in all the centuries thereafter. It is essential that all Catholics examine closely how they practice their faith. Any willful denial of doctrines of the Church is not Catholic and is grounds for removal from the Church.

Many Catholics are also unaware of the Church's teaching (above) pertaining to the possibility of heresy among Popes. If a true Pope were to accidentally try to teach error in faith and morals, the doctrine of Papal Infallibility teaches that this would be prevented by the Holy Ghost by such things as inspirations of grace, signs, intervention of other people, sickness or severe trial, and even an early death if necessary. Though if a Pope were to willfully teach error in faith or morals, that is when the Church states (as in the quotes above) that the Pope will be automatically labeled a heretic, lose his authority entirely, and automatically be cast from the Church.