Is there Catholic teaching and/or prophecy that a pope may be the false prophet of Revelation and the Church the whore of Babylon?

The short answer is that there is no teaching by the Roman Catholic church that any pope will be a false prophet, or that the church is the "who*e of babylon".

The long answer depends upon exactly what it means to be a catholic, and who gets to decide who is, or is not, appropriately so described. There are groups of people who claim to be catholic who believe that there is not now a legitimate pope so that the current claimants to the chair of Peter are not legitimate. As to how long it has been since there was a legitimate pope, I have seen different groups claim that the event happened after the death of different previous popes, but one of the more widespread claims is that the last legitimate Pope was Pius X. The followers of the late Archbishop LeFebvre in the SSPX fell into this group until recently. They, or others like they were, collectively known as sede vacantists, form a minority of those who call themselves catholic, but claim that subsequent Popes have embraced modernism, and become heretics, and are therefore no longer catholic, despite what they call themselves. Thus the situation where some who call themselves Catholics deny that the members of the current Roman Catholic Church are really catholic, and are instead heretics, and claim that the Popes are heretics, (and perhaps the Antichrist), and that what most consider to be the Roman Catholic Church is no longer so. The website referred to by the OP claiming that Pope Francis is the AntiChrist, appears to be a page authored by a person or persons holding or sympathetic to these views.

The commentary on 2 Thessalonians 2:4 (that Antichrist "sitteth in the temple of God") of the very anti-Protestant Original Douay-Rheims Bible says:

St. Augustine therefore li. 20 de civit. c. 19 and St. Jerome q. 11 ad Algasiam. do think, that this sitting of Antichrist in the temple, doth signify his sitting in the Church of Christ, rather than in Solomon's temple. Not as though he should be a chief member of the Church of Christ, or a special part of his body mystical, and be Antichrist and yet withal continuing within the Church of Christ, as the Heretics feign, to make the Pope Antichrist (whereby they plainly confess and agnise [recognize] that the Pope is a member of the Church, & in ipso sinu Ecclesia, and in the very bosom of the Church, say they:) for that is ridiculous, that all Heretics whom St. John calleth Antichrists as his precursors, should go out of the Church, and the great Antichrist himself should be of the Church, and in the Church, and continue in the same. And yet to them that make the whole Church in revolt from God, this is no absurdity. But the truth is, that this Antichristian revolt here spoken of, is from the Catholic Church: and Antichrist, if he ever were of or in the Church, shall be an Apostate and a renegade out of the Church, and he shall usurp upon it by tyranny, and by challenging worship, religion, and government thereof, so that himself shall be adored in all the Churches of the world which he list to leave standing for his honor. And this is to sit in the temple or *against the Temple of God, as some interpret. If any Pope did ever this, or shall do, then let the Adversaries call him Antichrist.

This passage makes it clear that a true pope cannot be Antichrist, but Antichrist will be an anti-pope, and anti-popes are not members of the Church.

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Cardinal Manning, the most prominent father at the First Vatican Council who was responsible for the definition on papal infallibility, said in his Temporal Power of the Vicar of Christ 2nd part (4 lectures), "The Perpetual Conflict of the Vicar of Christ", pp. 81-173 (available in printed form from TradiBooks as The Pope and The Antichrist):

We have here [2 Thessalonians 2:3-11] a prophecy … of a [spiritual*] revolt, which shall precede the second coming of our Lord … The authority, then, from which the revolt is to take place is that of the kingdom of God on earth, prophesied by Daniel [cf. Daniel 2] as the kingdom which the God of heaven should set up … in other words, the one and universal Church, founded by our Divine Lord, and spread by His Apostles throughout the world. In this one only kingdom was deposited the true and supernatural pure theism, or knowledge of God, and the true and only faith of God incarnate, with the doctrines and laws of grace. This, then, is the authority from which the revolt is to be made, be that revolt what it may.

[*"St. Jerome, with some others, interprets this revolt to be the rebellion of the nations or provinces against the Roman Empire. … They have revolted, and no manifestation has appeared." Thus, the revolt is spiritual, not temporal.]

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There's also Pope Leo XIII's Prayer to St. Michael the Archangel, which the Pope wrote after having a vision; its full version includes:

These most crafty enemies have filled and inebriated with gall and bitterness the Church, the spouse of the immaculate Lamb, and have laid impious hands on her most sacred possessions. In the Holy Place itself, where has been set up the See of the most holy Peter and the Chair of Truth for the light of the world, they have raised the throne of their abominable impiety, with the iniquitous design that when the Pastor has been struck, the sheep may be scattered.

(source: the 1910 The Raccolta or Collection of indulgenced prayers and good works, p. 125)

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cf. also The Permanent Instruction of the Alta Vendita: A Masonic Blueprint for the Subversion of the Catholic Church.

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From all these quotes, it is clear that Antichrist will be a pope who apostatizes from the Church and thus becomes an anti-pope.