The 'go to' text in the Bible for support of the Roman view of St. Peter is Matthew 16:18: 'And I also say to you that you are Peter, and on this rock I will build My church.' The claims of supreme universal jurisdiction are based on the interpretation that the 'rock' that Jesus refers to here is St. Peter himself. However, this view was a minority in the early Church. A 17th century Roman Catholic scholar, Jean de Launoy, spent a great deal of time researching the writings of the early Church Fathers in regards to their understanding of this passage, and found that there were 17 who believed that St. Peter was the 'rock', 44 who thought that the 'rock' was St. Peter's confession of faith, 16 who believed that the 'rock' was Christ Himself, and 8 who believed that the 'rock' was the apostles. So there was no early, historical consensus as to the meaning of this passage, but about 80% of the Fathers didn't consider the person of St. Peter to be the 'rock' referred to by Christ. In fact, this information was used by Archbishop Kenrick of St. Louis at the First Vatican Council to argue against the doctrine of papal infallibility, but clearly his argument did not prevail.
Even more than simply disagreeing amongst themselves, there is evidence of Church Fathers changing their own minds of the understanding of this passage. St. Augustine at points early in his career believed that St. Peter was the 'rock', and then at later points put forth that Christ was the 'rock':
'For on this very account the Lord said, "On this rock I will build my Church," because Peter had said, "Thou art the Christ, the Son of the living God." On this rock, therefore, He said, which thou hast confessed, I will build my Church. For the Rock (petra) was Christ' and on this foundation was Peter himself also built.' - Tractate CXXIV, On the Gospel of John, The Nicene and Post-Nicene Fathers, First Series, Vol. VII, p. 450
'Thou are therefore, saith He, Peter, and upon this Rock which thou hast confessed, upon this rock, which thou hast acknowledged, saying, "Thou art the Christ, the Son of the living God," I will build my Church. Upon Me I will build thee, not Me upon thee." - Sermon 76: 1. P.L. 38:479
Towards the end of his life, St. Augustine wrote the Retractions, wherein, with the benefit of maturity, he attempted to correct and refine what he thought to be wrong or misleading in his earlier works. He ruminates on his younger days, when he considered Peter to be the 'rock', and then goes on to explain:
'But I know that I have afterwards in very many places so expounded the Lord's saying, "Thou art Peter and upon this Rock I will build My Church" as to be understood of Him whom Peter confessed, when he said, "Thou art the Christ, the Son of the living God"...For it was not said to him, "Thou art the Rock (petra), but, Thou art Peter" (petrus). But Christ was the Rock, whom Simon confessing, as the Whole Church confesses Him, was called Peter. But of these two meanings let the reader choose the more probable.' - Retractiones, 1:21. P.L. 32:618
So, St. Augustine has changed his own mind, and, appears to not place any doctrinal significance on the question at all. As the author, Mr. Whelton puts it, St. Augustine 'really doesn't give two pins which one the reader chooses.'
To the point, most scholars agree that Matthew 16:18 holds no 'currency' as the basis for papal primacy in the early Church.
'But before the third century there was no call for a sustained, theoretical justification of this leadership. All were brethren, but the church in Rome was accepted as first among equals. The "Petrine text" of Matthew 16:18, "Thou art Peter and upon this rock I will build my Church," cannot be seen to have played any part in the story of Roman leadership and authority before the middle of the third century when the passionate disagreement between Cyprian of Carthage and Stephen of Rome about baptism apparently led Stephen to invoke the text as part of his defense against Cyprian. But it was not until Damasus in 382 that this Petrine text seriously began to become important as providing a theological and scriptural foundation on which claims to Primacy were based....From Damasus onwards there is a marked crescendo in the expression of the claims made by the bishops of Rome.' - Henry Chadwick, The Early Church, p. 237
Another quote from St. Augustine used to argue for papal supremacy is:
'the primacy of the Apostolic chair always existed in the Roman Church' - Ep. xliii., n.7)
However, the argument against this line is really very simple. It is claimed that the definite article 'the' is the incorrect translation. Instead, according to the editors of The Nicene and Post-Nicene Fathers, the quote should read, 'the primacy of an Apostolic chair always existed in the Roman Church.' Which quote, they say, is in line with the content of St. Augustine's other works, including the rest of the epistle that this quote is drawn from. For example, this paragraph from the same epistle:
'Well let us suppose that those bishops who decided the case at Rome were not good judges; there still remained a plenary Council of the Universal Church, in which these judges themselves might be put on their defense; so that, if they were convicted of a mistake, their decisions might be reversed.' - Confessions, op. cit., p. 282
Since one of the judges in question at this moment was Melchiades (aka Miltiades), Bishop of Rome, it is clear that St. Augustine sees the bishop of Rome, along with all his brother bishops, as being subject to the authority of a general council of the Church. In this, and elsewhere in his writings, I think it's fairly clear that St. Augustine holds the authority of councils of the whole Church to be greater in authority than any single bishop, including the bishop of Rome.
Another Augustinian quote used to support papal claims is:
'You are not to be looked upon as holding the true Catholic faith if you do not teach that the faith of Rome is to be held' - Sermon CXX n. 13
Here, one may argue, St. Augustine is saying that Rome is the arbiter of faith for the universal Church, and that anyone who deviates from her teaching lapses automatically and immediately into heresy. The author, Mr. Whelton, says, 'I spent countless hours looking through volumes of books for this Sermon CXX-nothing! At last I found a book written by Edward Denny, M.A., an Anglican scholar, in 1912...He comments on this quotation taken from Sermon CXX n. 13':
'The reference given is Sermon CXX N. 13. This quotation is not to be found in the sermon so numbered, or in any other of St. Augustine's in the Benedictine Edition. In a Sermon, however, which Cardinal Mai has published in his work, Patrum nova Bibliotheca, and attributed by him to St. Augustine, the words occur.
'Several reasons combine to suggest that the learned Cardinal is in error in considering this Sermon to be by St. Augustine: (a) First, this Sermon does not appear to have been attributed to St. Augustine by any other editor of his works. This is a weighty argument against his authorship thereof...(b) Secondly, the internal evidence afforded by the Sermon itself is against the alleged Augustinian authorship, as it manifestly deals with a condition of affairs in Africa which arose after St. Augustine's death in A.D. 431....written after the conquest of Rome in 455 by the Vandals.' - Paplism: A Treatise on the Claims of the Papacy As Set Forth in THE ENCYCLICAL SATIS COGNITUM
On this same theme, the author includes this quote from the index of The Faith of the Early Fathers by William A. Jurgens: 'Over the years numerous supplementary collections of sermons attributed to Augustine were published: 201...by Angelo Cardinal Mai, the larger part of which are regarded as spurious.' In other words, this sermon is not from St. Augustine, and therefore loses it's 'force'.
This quote from St. Cyprian is also used in papal claims: 'To be in communion with Cornelius is to be in communion with the Catholic Church' - Ep. Lv. n. 1 (I should like to note, in case there might be some confusion, that when the term 'Catholic Church' is applied, it is not in reference to the entity as it exists today, but rather in the sense of catholic as in 'universal' which is how the term was originally applied.)
This quotation is taken from a letter by St. Cyprian to Antonianus and addresses the problem of the latters' wavering loyalty between two contestants for the See of Rome: Novation and Cornelius. St. Cyprian is merely informing Antonianus that Cornelius is the rightful candidate. Cornelius had been consecrated by sixteen bishops, while Novation 'strives by bribery to be made an adulterous and extraneous bishop by the hands of deserters.' St. Cyprian goes on to tell him that to be in communion with a schismatic is to place oneself outside the Catholic Church. He urges Antonianus to recognize Bishop Cornelius, for it is only be continuing communion with canonical bishops that one remains in communion with the Church. Thus he advises, 'To be in communion with Cornelius is to be in communion with the Catholic Church.' St. Cyprian is not preaching communion with Cornelius because he is head of the universal Church, and makes this clear in the same letter: 'While the bond of concord remains, and the undivided sacrament of the Catholic Church endures, every bishop disposes and directs his own acts, and will have to give an account of his purposes to the Lord.' The independence of bishops in sacramental union with each other contains the totality of the universal Church. There is nothing in this letter that hints at any elevated status being awarded to Cornelius. St. Cyprian even refers to him as 'Cornelius our colleague.' The author includes several quotes from St. Cyprian showing his stance to be exactly opposite of papal supremacy, but I'll include only one:
'For neither does any of us set himself up as bishop of bishops, nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than himself can judge another. But let us all wait for the judgment of our Lord Jesus Christ, who is the only one that has the power of both preferring us in the government of His Church, and of judging us in our conduct there.' - Address to the Seventh Council of Carthage, The Ante-Nicene Fathers, p. 565
The next quote needs some background. Pope Celestine had excommunicated Nestorius, Patriarch of Constantinople for heresy - he denied that Mary is the mother of God, the Theotokos. However, the fact that he excommunicated a Patriarch shouldn't be read as though he enjoyed special privileges in that respect. Ignoring Celestine's excommunication of Nestorious, Emperor Theodosius called for a general council of the Church, which convened at Ephesus in 431 to settle the issue. The council Fathers disregarded Pope Celestine's excommunication, and treated Patriarch Nestorious with all the honor due his office, while the council was ongoing. The council proceeded to examine all the evidence, and to then pronounce its own sentence: 'The holy Synod...to Nestorious the new Judas: Know that for thy impious doctrines thou wast deposed by the holy Synod agreeably to the laws of the Church.' Not 'deposed by his holiness the Pope', but 'deposed by the holy Synod'. And prior to their decision, the Pope's declaration was ignored, entirely. So, the quote, from one Philip, a papal legate at the Council:
'Philip, the priest the Pontifical legate at the Council of Ephesus, no voice being raised in dissent, recalls: "No one can doubt, yea, it is known unto all ages, that St. Peter, the Prince of the Apostles, the pillar of the faith and the ground of the Catholic Church, received the keys of the Kingdom from our Lord Jesus Christ. That is: the power of forgiving and retaining sins was given to him who, up to the present time, lives and exercises judgment in the persons of his successors.'
The important bit, here, for Roman claims, is the 'no voice being raised in dissent'. It's sort of being read backwards, as though the Council was just assenting to the previous declaration of the Pope, when in fact, they came to their own conclusion, and were basically ignoring Pope Celestine's excommunication. So, not so much, there, I think.
Lastly, Pope Leo XIII claims that no evolution of papal supremacy has ever taken place:
'Wherefore, in the decree of the Vatican Council as to the nature and authority of the primacy of the Roman Pontiff, no newly conceived opinion is set forth, but the venerable and constant belief of every age.' - Sess. IV, cap. 3
However, this is at odds with both external and some internal scholarship on the matter. I include one last quote:
'As was recognized by both sides, the actual association of a Petrine ministry exclusively with the bishop of Rome was the result of a very gradual and complex development. For a long time, Catholics held a rather simple view: Jesus himself conferred primacy in the Church on Peter (three texts were most frequently cited in favor of this view: Matthew 16:17-19, Luke 22:32, and John 21:15-17), and Peter passed this primacy on to the bishop of Rome as his successor. But today Scripture scholars of all faiths read these texts to mean only that Jesus conferred a certain preeminence on Peter among the apostles but did not directly establish the papacy as a continuing office of the Church...
'There is no solid historical evidence that the bishop of Rome even claimed such a primacy until about the middle of the third century.' - Dynamic Catholicism: A Historical Catechism, Thomas Bokenkotter, p. 109