The excommunication of Pope Honorius I.

Yes, Popes can be heretics!  If it can happen "then," it can happen now.  From the Third Council of Constantinople:

The Sixth Ecumenical Council:
The holy and Ecumenical Synod further says, this pious and orthodox Creed of the Divine grace would be sufficient for the full knowledge and confirmation of the orthodox faith.  But as the author of evil, who, in the beginning, availed himself of the aid of the serpent, and by it brought the poison of death upon the human race, has not desisted, but in like manner now, having found suitable instruments for working out his will (we mean Theodorus, who was Bishop of Pharan, Sergius, Pyrrhus, Paul and Peter, who were Archbishops of this royal city, and moreover, Honorius who was Pope of the elder Rome, Cyrus Bishop of Alexandria, Macarius who was lately bishop of Antioch, and Stephen his disciple), has actively employed them in raising up for the whole Church the stumbling-blocks of one will and one operation in the two natures of Christ our true God, one of the Holy Trinity; thus disseminating, in novel terms, amongst the orthodox people, an heresy similar to the mad and wicked doctrine of the impious Apollinaris, Severus, and Themistius, and endeavouring craftily to destroy the perfection of the incarnation of the same our Lord Jesus Christ, our God, by blasphemously representing his flesh endowed with a rational soul as devoid of will or operation.  Christ, therefore, our God, has raised up our faithful Sovereign, a new David, having found him a man after his own heart, who as it is written, “has not suffered his eyes to sleep nor his eyelids to slumber,” until he has found a perfect declaration of orthodoxy by this p. 345 our God-collected and holy Synod; for, according to the sentence spoken of God, “Where two or three are gathered together in my name, there am I in the midst of them,” the present holy and Ecumenical Synod faithfully receiving and saluting with uplifted hands as well the suggestion which by the most holy and blessed Agatho, Pope of ancient Rome, was sent to our most pious and faithful Emperor Constantine, which rejected by name those who taught or preached one will and one operation in the dispensation of the incarnation of our Lord Jesus Christ who is our very God, has likewise adopted that other synodal suggestion which was sent by the Council holden under the same most holy Pope, composed of 125 Bishops, beloved of God, to his God-instructed tranquillity, as consonant to the holy Council of Chalcedon and to the Tome of the most holy and blessed Leo, Pope of the same old Rome, which was directed to St. Flavian, which also this Council called the Pillar of the right faith; and also agrees with the Synodal Epistles which were written by Blessed Cyril against the impious Nestorius and addressed to the Oriental Bishops.  Following the five holy Ecumenical Councils and the holy and approved Fathers, with one voice defining that our Lord Jesus Christ must be confessed to be very God and very man, one of the holy and consubstantial and life-giving Trinity, perfect in Deity and perfect in humanity, very God and very man, of a reasonable soul and human body subsisting; consubstantial with the Father as touching his Godhead and consubstantial with us as touching his manhood; in all things like unto us, sin only excepted; begotten of his Father before all ages according to his Godhead, but in these last days for us men and for our salvation made man of the Holy Ghost and of the Virgin Mary, strictly and properly the Mother of God according to the flesh; one and the same Christ our Lord the only-begotten Son of two natures unconfusedly, unchangeably, inseparably indivisibly to be recognized, the peculiarities of neither nature being lost by the union but rather the proprieties of each nature being preserved, concurring in one Person and in one subsistence, not parted or divided into two persons but one and the same only-begotten Son of God, the Word, our Lord Jesus Christ, according as the Prophets of old have taught us and as our Lord Jesus Christ himself hath instructed us, and the Creed of the holy Fathers hath delivered to us; defining all this we likewise declare that in him are two natural wills and two natural operations indivisibly, inconvertibly, inseparably, inconfusedly, according to the teaching of the holy Fathers.  And these two natural wills are not contrary the one to the other (God forbid!) as the impious heretics assert, but his human will follows and that not as resisting and reluctant, but rather as subject to his divine and omnipotent will.  For it was right that the flesh should be moved but subject to the divine will, according to the most wise Athanasius.  For as his flesh is called and is the flesh of God the Word, so also the natural will of his flesh is called and is the proper will of God the Word, as he himself says:  “I came down from heaven, not that I might do mine own will but the will of the Father which sent me!” where he calls his own will the will of his flesh, inasmuch as his flesh was also his own.  For as his most holy and immaculate animated flesh was not destroyed because it was deified but continued in its own state and nature (ὄρῳ τε καὶ λόγῳ), so also his human will, although deified, was not suppressed, but was rather preserved according to the saying of Gregory Theologus:  “His will [i.e., the Saviour’s] is not contrary to God but altogether deified.”

http://www.sacred-texts.com/chr/ecf/214/2140375.htm

Continuing:
Therefore we declare that in him there are two natural wills and two natural operations, proceeding commonly and without division:  but we cast out of the Church and rightly subject to anathema all superfluous novelties as well as their inventors:  to wit, Theodore of Pharan, Sergius and Paul, Pyrrhus, and Peter (who were archbishops of Constantinople), moreover Cyrus, who bore the priesthood of Alexandria, and with them Honorius, who was the ruler (πρόεδρον) of Rome, as he followed them in these things.  Besides these, with the best of cause we anathematize and depose Macarius, who was bishop of Antioch, and his disciple Stephen (or rather we should say master), who tried to defend the impiety of their predecessors, and in short stirred up the whole world, and by their pestilential letters and by their fraudulent institutions devastated multitudes in every direction.  Likewise also that old man Polychronius, with an infantile intelligence, who promised he would raise the dead and who when they did not rise, was laughed at; and all who have taught, or do teach, or shall presume p. 348 to teach one will and one operation in the incarnate Christ.…But the highest prince of the Apostles fought with us:  for we had on our side his imitator and the successor in his see, who also had set forth in his letter the mystery of the divine word (θεολογίας).  For the ancient city of Rome handed thee a confession of divine character, and a chart from the sunsetting raised up the day of dogmas, and made the darkness manifest, and Peter spoke through Agatho, and thou, O autocratic King, according to the divine decree, with the Omnipotent Sharer of thy throne, didst judge.

http://www.sacred-texts.com/chr/ecf/214/2140376.htm

Continuing:
Serious illnesses call for greater helps, as you know, most blessed [father]; and therefore Christ our true God, who is the creator and governing power of all things, gave a wise physician, namely your God-honoured sanctity, to drive away by force the contagion of heretical pestilence by the remedies of orthodoxy, and to give the strength of health to the members of the church.  Therefore to thee, as to the bishop of the first see of the Universal Church, we leave what must be done, since you willingly take for your standing ground the firm rock of the faith, as we know from having read your true confession in the letter sent by your fatherly beatitude to the most pious emperor:  and we acknowledge that this letter was divinely written (perscriptas) as by the Chief of the Apostles, and through it we have cast out the heretical sect of many errors which had recently sprung up, having been urged to making a decree by Constantine who divinely reigns, and wields a most clement sceptre.  And by his help we have overthrown the error of impiety, having as it were laid siege to the nefarious doctrine of the heretics.  And then tearing to pieces the foundations of their execrable heresy, and attacking them with spiritual and paternal arms, and confounding their tongues that they might not speak consistently with each other, we overturned the tower built up by these followers of this most impious heresy; and we slew them with anathema, as lapsed concerning the faith and as sinners, in the morning outside the camp of the tabernacle of God, that we may express ourselves after the manner of David, 334 in accordance with the sentence already given concerning them in your letter, and their names are these:  Theodore, bishop of Pharan, Sergius, Honorius, Cyrus, Paul, Pyrrhus and Peter.  Moreover, in addition to these, we justly subjected to the anathema of heretics those also who live in their impiety which they have received, or, to speak more accurately, in the impiety of these God-hated persons, Apollinaris, Severus and Themestius, to wit, Macarius, who was the bishop of the great city of Antioch (and him we also stripped deservedly of his pastor’s robes on account of his impenitence concerning the orthodox faith and his obstinate stubbornness), and Stephen, his disciple in craziness and his teacher in impiety, also Polychronius, who was inveterate in his heretical doctrines, thus answering to his name; and finally all those who impenitently have taught or do teach, or now hold or have held similar doctrines.

http://www.sacred-texts.com/chr/ecf/214/2140377.htm

Some useful commentary:

http://www.sacred-texts.com/chr/ecf/214/2140378.htm