What is the evidence that the New Testament authors believed, or did not believe, that they were writing an inspired record?

Paul directed in many places to both the tradition he transmitted to their fellow Christians, and to the Jewish Scriptures (which for him, as for the rest of the disciples and for Jesus meant the Septuagint). For instance, in 2 Timothy 3:14–17, Paul says:

But as for you, continue in what you have learned and have firmly believed, knowing from whom you learned it, and how from childhood you have been acquainted with the sacred writings which are able to instruct you for salvation through faith in Christ Jesus. All Scripture is inspired by God and profitable for teaching, for reproof, for correction, and for training in righteousness; so that the man of God may be equipped, prepared for every good work.

So, in terms of the term "Scripture" above, it is unlikely Paul is referring to his own letters (let alone to the gospels, which might have not been written by then). Most likely. Cardinal Newman states regarding this:

"Now, a good part of the New Testament was not written in his boyhood: Some of the Catholic epistles were not written even when Paul wrote this, and none of the books of the New Testament were then placed on the canon of the Scripture books. He refers, then, to the scriptures of the Old Testament."

We can find a more positive side on John. It was already noted in another answer that Revelation is quite vocal about the authority of truth in the book. Something similar can be found in John 20:30-31, which states:

Jesus performed many other signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Messiah, the Son of God, and that by believing you may have life in his name.

And let's not forget what John had written earlier, in 14:26:

But the Advocate, the Holy Spirit, whom the Father will send in my name, will teach you all things and will remind you of everything I have said to you.

Thus, John is not only "writing the truth", but he is probably aware that the Holy Spirit is behind such effort.

To understand the question, one must define the word "inspired".

outstanding or brilliant in a way or to a degree suggestive of divine inspiration https://www.merriam-webster.com/dictionary/inspired

This word "inspiration" now begs for additional definition.

a divine influence or action on a person believed to qualify him or her to receive and communicate sacred revelation https://www.merriam-webster.com/dictionary/inspiration

So, is there evidence the New Testament writers thought their writings were "inspired of God to communicate sacred information"? The answer of course is yes.

To introduce this, we first need to consider whether they thought the Old Testament was inspired. I suspect everyone agrees that they would have thought yes. The phrase "thus saith the LORD" is the typical formula. It is a divine signature as it were. They speak of God meeting Moses. They speak of the LORD prophesying. They speak the phrase some 415 times in the King James Version.

With that in mind, do we find a similar phrase in the New Testament? If so, with the old phrase indicating divine inspiration, what would its fulfillment indicate, if not the exact same thing?

Now all this was done, that it might be fulfilled which was spoken of the Lord by the prophet, saying, Mat 1:22

But those things, which God before had shewed by the mouth of all his prophets, that Christ should suffer, he hath so fulfilled. Acts 3:18ph

Why would they use this divine building block of thus saith the LORD with it is inspired fulfilled?

But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name. John 20:31

So yes, the New Testament writers knew they were writing an inspired record.

The four New Testament gospels were all written anonymously, so in spite of second-century attributions to Matthew, Mark, Luke and John, we do not really know who wrote them and therefore they can not tell us whether they believed their gospels were divinely inspired. The gospels, as a genre, are narrated by third-party, omniscient narrators who are portrayed as knowing every private action, thought and inner feeling of the participants. Rhoads, Dewey and Michie say, in Mark as Story, third edition page 43, the narrator of Mark gives the audience privileged knowledge. One explanation could be that the authors wrote in this way because of divine inspiration that gave them these insights, although other explanations are also possible.

The author of Acts of the Apostles seems to have been inspired by the ancient play, the Bacchae by Euripides (d. 406 BCE) in Paul's conversion on the road to Damascus, and in the story of Paul's escape from prison, both of which have close parallels to scenes in the play. Because of this and his probable reliance on the works of Josephus (on this, see Richard Carrier's article on Luke and Josephus, where Carrier concludes Luke almost certainly knew and drew upon the works of Josephus), it is difficult to believe he also felt himself to be writing under divine guidance.

Whether or not Jesus' disciple Peter wrote the letter now known as Second Peter, it is strong evidence that Paul's epistles were seen as Scripture at a very early stage in Christianity. This need not mean that Paul himself thought of his epistles as divinely inspired. Paul certainly intended his readers to take his letters as authoritative, but there is nothing in the texts to suggest that he believed he was writing under divine inspiration. Christopher D. Stanley says, in As It Is Written, page 3, that the wording of Paul's quotations and allusions from the Hebrew scriptures often diverges significantly from the original wording, and that he also deviates fairly often from what modern readers might see as the "original meaning" of the passages to which he refers. This seems incompatible with a sense of divine inspiration.

Some of the New Testament epistles have been judged by many critical scholars as pseudepigraphical, which if true is hardly consistent with a sense of divine inspiration. For example, a clear majority of scholars believe the 'Pastoral Epistles' (Titus, 1 Timothy, 2 Timothy) were written early in the second century, and that the authors used Paul's name to have their message more readily accepted. Most scholars would place 2 Peter even later, pointing out that it uses material from The Epistle of Jude, which inadvertently self-identifies as a second-century epistle.

The question is:

Did the NT writers themselves know that they were writing an inspired record?

Though there is at least one major exception, which is mentioned below, for the most part the New Testament authors simply wrote their accounts without commenting on whether they were inspired. We can therefore glean only a few clues here and there about their state of mind, and in particular, about their sense of whether they were writing on their own or under inspiration from God.

Here are a few such clues contained within several of the books in the New Testament.

Luke and Acts

This is the often overlooked introduction to the Gospel of Luke:

Inasmuch as many have undertaken to compile a narrative of the things that have been accomplished among us, just as those who from the beginning were eyewitnesses and ministers of the word have delivered them to us, it seemed good to me also, having followed all things closely for some time past, to write an orderly account for you, most excellent Theophilus, that you may have certainty concerning the things you have been taught. (Luke 1:1-4)

And the parallel introduction to the book of Acts:

In the first book, O Theophilus, I have dealt with all that Jesus began to do and teach, until the day when he was taken up, after he had given commands through the Holy Spirit to the apostles whom he had chosen. He presented himself alive to them after his suffering by many proofs, appearing to them during forty days and speaking about the kingdom of God. (Acts 1:1-3)

Here we find the unidentified (in the text itself) author of these two books speaking confidently of "having followed all things closely for some time past," (KJV: "having had perfect understanding of all things from the very first"), and citing this as his basis for writing "an orderly account" of these events.

Clearly the author of the Gospel of Luke felt that he was writing an account based on his own knowledge and investigation into the subjects and events he wrote about. This strongly suggests that the author of Luke and Acts did not think of himself as being divinely inspired in the writing process.

Paul

Much material could be pulled out of Paul's letters to attest to his state of mind in writing. However, here is one particularly clear passage about the author's sense of the source of his writing:

To the married I give this charge (not I, but the Lord): the wife should not separate from her husband. . . . To the rest I say (I, not the Lord) that if any brother has a wife who is an unbeliever, and she consents to live with him, he should not divorce her. (1 Corinthians 7:10, 12, italics added)

Here Paul, the author identified in 1 Corinthians 1:1, distinguishes between:

1. "charges" or commands, given not by himself, but by the Lord, and
2. instructions given not by the Lord, but by himself.

This is a clear indication that the author believed that at least some of what he wrote was on his own initiative and from his own belief and understanding rather than being inspired by the Lord.

Revelation

By contrast, the book of Revelation starts out with this prologue:

The revelation of Jesus Christ, which God gave him to show to his servants the things that must soon take place. He made it known by sending his angel to his servant John, who bore witness to the word of God and to the testimony of Jesus Christ, even to all that he saw. Blessed is the one who reads aloud the words of this prophecy, and blessed are those who hear, and who keep what is written in it, for the time is near. (Revelation 1:1-3)

The book then speaks in the voice of its human author:

John to the seven churches that are in Asia: (Revelation 1:4)

Here we find the human author clearly proclaiming this book as a "revelation of Jesus Christ," which was made known "by sending his angel to his servant John."

Clearly the author saw himself as writing under divine inspiration through the mediation of an angel. And indeed, a number of times he indicates that he was in an inspired, or spiritual state, had heaven opened to him, and heard voices from heaven. For example:

I was in the Spirit on the Lord's day, and I heard behind me a loud voice like a trumpet. saying, "Write what you see in a book and send it to the seven churches, to Ephesus and to Smyrna and to Pergamum and to Thyatira and to Sardis and to Philadelphia and to Laodicea." (Revelation 1:10-11)

After this I looked, and behold, a door standing open in heaven! And the first voice, which I had heard speaking to me like a trumpet, said, "Come up here, and I will show you what must take place after this." At once I was in the Spirit, and behold, a throne stood in heaven, with one seated on the throne. (Revelation 4:1-2)

And when the seven thunders had sounded, I was about to write, but I heard a voice from heaven saying, "Seal up what the seven thunders have said, and do not write it down." (Revelation 10:4)

Then the voice that I had heard from heaven spoke to me again, saying, "Go, take the scroll that is open in the hand of the angel who is standing on the sea and on the land." (Revelation 10:8)

Then God's temple in heaven was opened, and the ark of his covenant was seen within his temple. There were flashes of lightning, rumblings, peals of thunder, an earthquake, and heavy hail. (Revelation 11:19)

These and other expressions in the book of Revelation indicate that its author felt a high level of spiritual and divine inspiration for the visions he recorded in the book.

Conclusion

This is by no means an exhaustive list of passages in the New Testament indicating something about the state of mind of the authors with regard to their sense of whether or not they were writing an inspired record.

However, even these few passages show that there was a wide variety of experiences on the part of the various New Testament authors, ranging from a sense of writing their books or letters based on their own ideas and efforts in some instances, all the way to a sense, in other instances, of receiving and recording words and visions given to them by God through the angels in heaven.