There is no biblical argument against Papal succession.
The authority of Peter to govern the Church is based on the words of Christ in Matthew 16:18
You are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven. [NIV]
Other verses point to Christ's headship and the common work of all the apostles, as in 1 Corinthians 3:11, Ephesians 2:20, 1 Peter 2:4–6, Rev 21:14 [all NIV]:
For no-one can lay any foundation other the one already laid, which is Jesus Christ.
...built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone.
As you come to him, the living Stone — rejected by men but chosen by God and precious to him — you also, like living stones, are being built into a spiritual house to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ. For in Scripture it says, "See, I lay a stone in Zion, a chosen and precious cornerstone, and the one who trusts in him will never be put to shame."
The walls of the city had twelve foundations, and on them were the names of the twelve apostles of the Lamb.
These all point to the basis of the Church's faith being faith in Christ, and the contribution of all the apostles.
Nevertheless, the dominical words which set Peter up as the rock on which the earthly church stands, and who has the power and authority to bind and loose on earth what is bound and loosed in heaven, remain.
The Church Fathers, those Christians closest to the apostles in time, culture, and theological background, clearly understood that Jesus promised to build the Church on Peter, as shown by their writings:
[T]he Lord said to Peter, ‘On this rock I will build my Church, I have given you the keys of the kingdom of heaven [and] whatever you shall have bound or loosed on earth will be bound or loosed in heaven’ [Matt. 16:18–19] ... What kind of man are you, subverting and changing what was the manifest intent of the Lord when he conferred this personally upon Peter? Upon you, he says, I will build my Church; and I will give to you the keys. (Tertullian, Modesty 21:9–10, AD220)
Look at [Peter], the great foundation of the Church, that most solid of rocks, upon whom Christ built the Church [Matt. 16:18]. And what does our Lord say to him? ‘Oh you of little faith,’ he says, ‘why do you doubt?’ [Matt. 14:31]. (Origen, Homilies on Exodus 5:4, AD248)
There is one God and one Christ, and one Church, and one chair founded on Peter by the word of the Lord. It is not possible to set up another altar or for there to be another priesthood besides that one altar and thata one priesthood. Whoever has gathered elsewhere is scattering. (Cyprian of Carthage, Letters 43:5, AD253)
You cannot deny that you are aware that in the city of Rome the episcopal chair was given first to Peter; the chair in which Peter sat, the same who was head—that is why he is also called Cephas [‘Rock’]—of all the apostles; the one chair in which unity is maintained by all. (Optatus, The Schism of the Donatists 2:2, AD367)
Philip, the presbyter and legate of the Apostolic See [Rome], said: ‘There is no doubt, and in fact it has been known in all ages, that the holy and most blessed Peter, prince and head of the apostles, pillar of the faith, and foundation of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the Savior and Redeemer of the human race, and that to him was given the power of loosing and binding sins: who down even to today and forever both lives and judges in his successors’. (Council of Ephesus, Acts of the Council, session 3, AD431)
Wherefore the most holy and blessed Leo, archbishop of the great and elder Rome, through us, and through this present most holy synod, together with the thrice blessed and all-glorious Peter the apostle, who is the rock and foundation of the Catholic Church, and the foundation of the orthodox faith, has stripped him [Dioscorus] of the episcopate. (Council of Chalcedon, Acts of the Council, session 3, AD451).
The last two quotes, from pre-Schism Ecumenical Councils, are significant. The New Testament canon was fixed between AD367 and 405, so the Councils in 431 and 451 would have been particularly aware of what was biblical and what was not.
Chalcedon calls Leo the Great "most holy and blessed, archbishop of the great and elder Rome," and lists no other members of the Council in that snippet; this puts the Pope in a position of authority. Ephesus arguably went further, agreeing that "Peter, head of the apostles, even to today and forever both lives and judges in his successors."
Thus the early Church prior to the Great Schism agrees that Peter's authority subsists in his successors in Rome.
None of this denies the trouble which led to the Great Schism and the Reformation; it seeks to answer the question as stated, "What is the biblical argument against Papal Succession?" with "There is none."