When was the Bible formally declared to be inerrant and/or infallible?

I'll answer the OP by addressing the why. The when would have corresponded to the why answer.

Inerrant and infallible mean incapable of being wrong.

The only thing who is always never wrong is God. How do we know something is of God? He sent prophets (OT) and apostles (NT).

The Muratorium Fragment makes this distinction between God's word and man's opinions, however pius or proclaimed they may be.

But Hermas wrote the Shepherd (74) very recently, [7c] in our times, in the city of Rome, (75) while bishop Pius, his brother, was occupying the [episcopal] chair (76) of the church of the city of Rome. [7d] (77) And therefore it ought indeed to be read; but (78) it cannot be read publicly to the people in church either among (79) the Prophets, whose number is complete, [8] or among (80) the Apostles, for it is after [their] time. MF

The prophets (OT) cannot be added to because their time was over (complete). It is the same idea circa 150 CE that the apostolic authority is also over (complete). In both instances, the canon of OT and NT was complete.

To be sure, there was ongoing argument about what that canon may be, but the why is clearly defined.

There are many religions that teach from other books, other traditions, other definitions, but for the early church the why is complete. It is only the word of God written in the OT and NT that is inerrant/infallible. All else falls short.

When was the Bible formally declared to be inerrant and/or infallible?

Many people and even denominations have declared the Bible inerrant. Most would clarify by saying "as originally written in the original language". Most of these declarations are relatively recent and largely in a defensive response to the massive assault on Biblical reliability by so-called scholars using textual or higher criticism, German rationalism, and a host of other techniques flowing from humanism and secularism originating in the ironically named period of enlightenment.

Prior to this period the Bible was assumed to be inerrant by almost everyone. Those who took a closer look (not many) might have acknowledged the superiority of one translation over another, but most would have not questioned that the Bible was truth as Jesus said;

John 17:17 Sanctify them through thy truth: thy word is truth.

There were of course many people who were not Christian and they mostly ignored the Bible.

The use of the phrase "formally declared" implies an authority that supersedes the Bible such that this authority in itself becomes the standard of truth. Some denominations may declare this authority and many "scholars" have implied in their declarations that they see themselves as having this authority. However, I see the Bible as declaring truth and that those who are of the truth will respond and those who are not are not able to understand.

1 Corinthians 2:14 But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.

The determination of inerrancy has to be in the hands of each person. Those called by God can judge most accurately. Those hostile to God may attack the reliability of the Bible with various assertions. Those weaker that God calls may be harmed by those who attack. Therefore those who attack should be warned;

Matthew 18:6 But whoso shall offend one of these little ones which believe in me, it were better for him that a millstone were hanged about his neck, and that he were drowned in the depth of the sea.

Here's some verses on infallibility:

• 2 Timothy 3:16
• Isaiah 55:10-11
• 2 Peter 1:19-21

The latter is worth citing here:

And we have something more sure, the prophetic word, to which you will do well to pay attention as to a lamp shining in a dark place, until the day dawns and the morning star rises in your hearts, 20knowing this first of all, that no prophecy of Scripture comes from someone’s own interpretation. 21For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit

Lest the reader shout circularity, the train of thought here is as follows:

1. Let us assume/trust the general historicity of Scripture (all we need to assume is also corroborated by seconday/tertiary accounts)
2. Then we know Peter is an apostle of Jesus, invested with great authority from him (able to call Paul's writings Scripture, 2 Peter 3:16)
3. So when Peter declares the infallibility of Scripture, which his readers would have understood to be all of the OT as well as Paul's writings, he is, right then and there, formally declaring that the Word of God through these writings is infallible.

To get back to your question, I can't recall a specific OT passage declaring its own infallibility, but there are plenty of passages that emphasize that God spoke directly to the writers of these books (Prophets basically just transcribed his words, for example) and I think that speaks for itself.

I've got to run but I'll examine more closely 2 Tim. 3:16 when I return and I'll also dig into my church history books. I know I've failed to meet some of your more strict answer requirements so I'll try to clean that up as well. In the meantime, feel free to edit this, folks.

Within various scripture texts can be found declarations that, because the written words are God's words to humanity, they are inspired of God, who cannot lie (Titus 1:2 & Hebrews 6:18). A few examples:

"...that he might make thee know that man doth not live by bread only, but by every word that proceedeth out of the mouth of the Lord doth man live." Deuteronomy 8:3 A.V.

Jesus quoted those very words to thwart the Devil's temptation to disbelieve what God had said. The Devil was trying to get Jesus to disbelieve God having declared him to be his beloved Son, only 40 days previously. Had Jesus begun to doubt the very words of God to him, personally, he would then have disobeyed God by doing what the Devil suggested. The Devil tried that tactic a second time, and (a second time) Jesus defeated his corrupted use of Old Testament scripture by saying "It is written..." For the third temptation, Jesus retorted, "Get thee hence, Satan: for it is written..." (Matthew 4:1-10)

Notice how the very words of God are written down? This was before any of the New Testament scriptures were written. Yet Jesus verified the truth and purity of that ancient written word, using it to live by the words of God, which is life. Isaiah had earlier shown the same fact. He wrote:

"...the word of our God shall stand forever." "For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts. For as the rain cometh down, and the snow from heaven, and returneth not thither, but watereth the earth, and maketh it bring forth and bud, that it may give seed to the sower and bread to the eater: so shall my word be that goeth forth out of my mouth: it shall not return unto me void, but it shall accomplish that which I please, and it shall prosper in the thing whereto I sent it." Isaiah 40:8 & 55:9-11 A.V.

Jesus also said, "Heaven and earth shall pass away, but my words shall not pass away." And, "Is it not written in your law... [quoting Psalm 82:6] ...unto whom the word of God came, and the scripture cannot be broken." Matthew 24:35 & John 10:34-35 A.V.

By calling that Psalm unbreakable, Jesus called the whole of the Hebrew Scriptures the word of God, which cannot be broken; the entirety of them are infallible - unbreakable. Men might try to break them, by disbelief or distortion, but God's words stand fast to eternity, and those words are the very words the Holy Spirit inspired some 40 men to write, over a period 1,600 years, to form a harmonious whole. Nothing has erased them, and for those who believe in the divine preservation of Scripture (as well as their divine inspiration), their faith is that which Paul spoke of here:

"So then faith cometh by hearing, and hearing by the word of God." Romans 10:17 A.V.

Hearing God's words verbally though preaching the gospel, or reading those words in scripture - either way, God uses his words to impart life. We have those words of God and Christ written down throughout the entire Holy Bible, and all the writers testify to that, living their lives by the written and spoken word of God. We have that written word to this day, and right from the earliest times of the Church, its leaders have vouched for its utter truthfulness.

But you ask for the time of "a formal declaration". Some examples of such statements from the end of the first century have already been given in an answer here. You also ask "why was inerrancy/infallibility formally declared?" That was due to rapid attacks by disbelievers or even those who claimed to believe the truth of scripture but who twisted what it said to come up with false doctrines. Even before the canon of scripture was closed, some of the New Testament apostles had written down warnings about false teachings threatening to lead Christians away from what the scriptures said about Christ and the gospel (the Gnostics, for instance - 1 John 3:11 & 4:1-3.) And Jude spoke of wicked men having crept into gatherings, to lead others astray (for financial gain and for carnal lusts - Jude verses 4 to 19.)

Once doubts are cast on the divine inspiration of Scripture, ("Hath God really said...?") it's not long till people think that the word of God is contained in Scripture, but that not all of it is actually the word of God. Then comes the inevitable doubting whether even some of the things Jesus is recorded as saying are factual.

"As B.B. Warfield argued powerfully against Marcus Dods [1834-1909], if we do not receive the Christ who is made known to us in Scripture - the Christ who said of Scripture, 'It cannot be broken' - then the alternative is another Christ, or, indeed, many Christs, all modified according to personal opinion." Evangelical Holiness, p.56 Iain H. Murray, The Banner of Truth Trust, 2013

A biblical case for inerrancy can certainly be made on the basis of passages like 2 Timothy 3:16 and 2 Peter 3:16, but these are limited in that they do not obviously refer to the entire Bible, both Old and New Testaments. So to address this question it's best to turn to the writings of the early fathers.

Clear statements of the truthfulness and perfection of the Bible appear very early. Here are several examples:

• Clement of Rome (?–99)

  Look carefully into the Scriptures, which are the true utterances of the Holy Spirit. Observe that nothing of an unjust or counterfeit character is written in them. (Epistle to the Corinthians, chapter 45. In chapter 47 he cites Paul as an inspired author)

• Justin Martyr (100–165)

  I am entirely convinced that no Scripture contradicts another (Dialogue with Trypho, chapter 65)

• Irenaeus (?–202)

  the Scriptures are indeed perfect, since they were spoken by the Word of God and His Spirit (Against Heresies, 2.28.2)

• Tertullian (155–240)

  The statements, however, of holy Scripture will never be discordant with truth. (A Treatise on the Soul, chapter 21)

The definition of "Scripture" for these authors varied to some extent, but later fathers, like Augustine, more carefully emphasized the importance of defining a canon and gave even more explicit statements of inerrancy:

I have learned to yield this respect and honour only to the canonical books of Scripture: of these alone do I most firmly believe that the authors were completely free from error. (Letter 82)