OP: what are the biblical arguments (Protestant or Orthodox) against the papacy and papal succession?
The papacy is the idea of "the ecclesiastical system in which the pope as successor of St. Peter and Vicar of Jesus Christ governs the Catholic Church as its supreme head." -New Advent-
There are at least two biblical arguments against this idea that have yet to be mentioned in this thread, but let me summarize the two main arguments already mentioned.
The "on this rock I will build my church" in Matthew 16 must refer to Peter's confession, not to Peter himself. I will detail the reason for this shortly, but suffice to say here that Christ says Christ Himself will build His church, not Peter and certainly not his so-called successors.
And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. Mat. 16:18
The "priesthood of all believers" is also a strong argument against the papacy. To look further into this would require developing the finality of Christ's one sacrifice once.
But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God; Heb. 10:12
But back to the two other arguments.
If Christ had actually conferred to Peter some so-called succession rights, then one would think Peter would know this. To a degree, he does. As Tertullian noted, Peter did exercise the keys by opening the kingdom doors to Jew and Gentile alike. As Christ said, the gates of hell will not prevail. It won't be because of men that this is true. Again as Christ said, Christ will build His church.
But did Peter confer something to Linus, the second Pope so-called? There is no record of this. Even the letter of Clement only mentions that he believes Peter died in Rome. There is no mention of something that supposedly is so important about some lineage of Popes. Besides, what we do have is Peter actually conferring Jesus' instructions to the elders of Asia Minor, not to the bishops in Rome.
Peter, an apostle of Jesus Christ, to the strangers scattered throughout Pontus, Galatia, Cappadocia, Asia, and Bithynia, … 1 Peter 1:1
After addressing his letter to all, Peter then narrows the focus.
The elders which are among you I exhort, who am also an elder, and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed: 1 Peter 5:1
And then Peter gives the elders (plural, not just one, but many) the same instructions.
Feed the flock of God which is among you, taking the oversight thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind; 1 Peter 5:2
He saith unto him the third time, Simon, son of Jonas, lovest thou me? Peter was grieved because he said unto him the third time, Lovest thou me? And he said unto him, Lord, thou knowest all things; thou knowest that I love thee. Jesus saith unto him, Feed my sheep. John 21:17
So, did Jesus give Peter a papal lineage? No, it was Peter's confession of faith, but even if you think Jesus did, Peter then gives it to the unknown elders in Asia Minor.
The first and last apostles to die were James and John, sons of Zebedee, sons of thunder. John writes a letter to Gaius called 3 John. In it he mentions a person, although some say it is a metaphor, a warning, called Diotrephes.
I wrote unto the church: but Diotrephes, who loveth to have the preeminence among them, receiveth us not. 3 John 1:9
This is written circa 100ad. The word "preeminence" is just as it is translated.
philopróteuó: to strive to be first
Original Word: φιλοπρωτεύω
Part of Speech: Verb
Phonetic Spelling: (fil-op-rote-yoo'-o)
Definition: to strive to be first
Usage: I love the chief place, desire preeminence.
This is a person in the church who wants to be first, to be in charge, to lead all the others even to the point of rejecting apostolic authority. This, of course, is another argument already mentioned (scripture as the rule of faith; wherein the New Testament is written between the sons of thunder as the valid apostolic testimony).
Paul uses a similar word when describing Christ as having the preeminence.
And he is the head of the body, the church: who is the beginning, the firstborn from the dead; that in all things he might have the preeminence. Col 1:18
The idea of a single individual or a lineage type of group having the preeminence is a very old idea, but one that is reserved for Christ alone who builds His church, not some one who loves to be first. John warns against a Diotrephes. Tertullian later picks up the same idea, rejecting it likewise, as usurping what belongs to Christ alone.
Moreover Peter, if he was given some authority with the keys, opened the kingdom to Jew and Gentile. The gates of hell will not prevail. Peter then gave that instruction to the elders in Asia Minor. Feed His sheep.