What is the biblical basis for the argument against the Catholic teaching on the primacy of Peter? The Catholic Church uses, for example, the following NT passages in support of its teaching on the primacy of Peter: 


Mt 16:13-20.
Lk 22-31-34.
Jn 21:15-23.


What is the biblical basis that argues against these passages as supporting the Catholic teaching on the primacy of Peter, and what others argue against the aforementioned Catholic teaching?

Related: What are the biblical arguments against the papacy and papal succession?



Background: This question was prompted by the comments exchange following this bruisedreed's answer.
The term 'Primacy of Peter' is primarily a Catholic term and comes highly loaded - in it's normal usage, it is inextricably linked to and fundamentally inseparable from the institution of the Papacy. It is worth noting that the original meaning of Primate is:


  one first in authority or rank (source, emphasis added)


While Protestants generally do not deny that Peter was a principal leader in the early church and that Christ specifically gave him responsibility to shepherd the flock, for them to use the term 'Primacy of Peter' to describe this - given the Catholic use of the term - would be highly unusual, and probably a signal that they are looking to leave their current church polity, and submit themselves to the authority of the Bishop of Rome. 

The Protestant view regarding the limitations of Peter's role really does hinge on the interpretation of the Matthew passage. Clearly a 'Rock' on which the Church will be built carries immense significance. As important as Peter was in the life of the early church, he eventually died and his legacy in terms of contribution to scriptural canon and role in establishing, nurturing and leading the early churches, seems much on a par with James ("the just", author of the epistle that bears his name, the Bishop of Jerusalem according to tradition, the "brother" of the Lord, not to be confused with either of the two James who were amongst the 12 disciples) and the Apostle John, and significantly less than that left by the Apostle Paul. From this, (ie with the benefit of hindsight) it doesn't seem that the significant terminology of the 'Rock on which I will build my church' accurately describes Peter's personal effect and ministry towards the early church: If the Rock is so important, it either must be pointing beyond the man to the actual role he supposedly embodied (ie a papal system of some kind), or be about something else again.

The case that the latter option should be preferred, is quite strong in consideration of the principle 'scripture interprets scripture': ie if we examine all the scriptures that speak to ecclesiological issues, do we see such a pivotal role as a pope (or a one-of-a-kind primate in Peter) given an important emphais? The answer is an emphatic no!

I will not reproduce all relevant scriptures, but will refer to one of the most significant in terms of this particular argument:


  19 Consequently, you are no longer foreigners and strangers, but fellow citizens with God’s people and also members of his household, 20 built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone. 21 In him the whole building is joined together and rises to become a holy temple in the Lord. - Ephesians 2:19-21 NIV


This is clearly taking about the church and it's establishment. In my view, the only way for someone to interpret this particular passage as being consistent with a papacy or even 'the Primacy of Peter' is to employ eisegesis to twist the scripture. On the other hand, if we use this passage to help interpret Matthew 16:18 particularly noting that the Prophets gave revelation concerning the Christ to come, while the Apostles bore witness to him and using 'Christ as cornerstone' as a key to guide us, the interpretation of the confession of Jesus as the Christ being what is referred to as the rock is eminently preferable. In fact, this is the only valid choice to be made if we are operating on the basis of harmonizing scripture.

A final scripture reference to refute the concept of a Primate as being compatible with biblical ecclesiology:


  25 Jesus called them together and said, “You know that the rulers of the Gentiles lord it over them, and their high officials exercise authority over them. 26 Not so with you. Instead, whoever wants to become great among you must be your servant, 27 and whoever wants to be first must be your slave— 28 just as the Son of Man did not come to be served, but to serve, and to give his life as a ransom for many.” - Matthew 20:25-28 NIV


As soon as you start naming someone as a 'Primate', you are transgressing this instruction, which is why the early Bishops of Rome were loathe to adopt the formula regarding their office that was used by later successors:


  "I say it without the least hesitation, whoever calls himself the universal bishop, or desires this title, is, by his pride, the precursor of Antichrist, because he thus attempts to raise himself above the others. The error into which he falls springs from pride equal to that of Antichrist; for as that Wicked One wished to be regarded as exalted above other men, like a god, so likewise whoever would be called sole bishop exalteth himself above others....You know it, my brother; hath not the venerable Council of Chalcedon conferred the honorary title of 'universal' upon the bishops of this Apostolic See [Rome], whereof I am, by God's will, the servant? And yet none of us hath permitted this title to be given to him; none hath assumed this bold title, lest by assuming a special distinction in the dignity of the episcopate, we should seem to refuse it to all the brethren." - Pope Gregory 'the Great' - see related question.