The temple was destroyed sometime around AD 70.
Which New Testament books were written after that catastrophe?
Do these books comment at all on the destruction of the temple?
The dating of the New Testament is a matter of some controversy, so there is no consensus answer to the question of which ones were written after the destruction of the temple. (The Old Testament books were of course written well before).
No New testament documents make clear reference to the destruction of the temple. Some appear to prophecy the destruction, but it is clearly written as prophecy and not fulfilment, and even then it is not an absolute certainty. Many scholars use this information to date the NT canon as fixed before the destruction of the temple, on the reasonable grounds that had it been known about some NT writers would certainly have mentioned it as evidence of the correctness of prophecy. Bishop John A Robinson makes a well-argued case for this in his book "Redating the New Testament", or its popular version "Can we Trust the New Testament?".
Extensive quotes of the New Testament in letters known to be written at the end of the 1st century AD show that the NT documents were well known and accepted by that time, although it is technically possible that they may have undergone further editing between then and the dates of the first positively dated manuscripts from a hundred years later.
So to answer the questions:
As others have noted, dates of the writing of New Testament books are disputed.
Let me use F. F. Bruce's "The New Testament Documents" as my major source here. This page -- http://www.freebeginning.com/new_testament_dates/ -- also gives dates, though some of these are rather earlier than the dates I've usually heard.
Without getting into all the arguments surrounding the dispute, I think most scholars put Mark as the first Gospel, circa AD 60-65, which would of course be before the destruction of the temple. The other Gospels are generally dated Luke 80-85, Matthew 85-90, and John 90-100. I think few would push Matthew and John far enough forward to be before the destruction of the temple. Bruce and others would date Luke before that, so that's the only one that I think is really questionable if it's before or after.
Acts is a "sequel" to Luke, and so must have been written after. (A few scholars say that the book that Acts follows is not the Gospel of Luke that we have but some other lost book, and then date it earlier.)
Bruce puts the writing of Paul's epistles between AD 48 and AD 64. I think most scholars date Paul's epistles before the Gospels.
The epistle of James is generally placed in the 50s.
Revelation and John's epistles are usually put about the same time as John's Gospel, AD 90-100.
So, books almost certainly written after the fall of Jerusalem: Gospel of John, the three epistles of John, and Revelation. Probably after: Luke and Acts. Very hard to say: Matthew and Jude. All the rest are probably before.
As to whether any books discuss the destruction of the temple: Well, Jesus clearly prophesied it, Matthew 24. Some preterists interpret Revelation 11 as referring to the destruction of the temple -- some date Revelation very early and see this as a prophecy, others see it as describing a past event. To the best of my knowledge, that's it.
Raymond E. Brown says in An Introduction to the New Testament, page 164, there is wide scholarly agreement that Mark's Gospel was written in the late 60s or just after 70, and therefore the destruction of the temple was imminent or had already occurred. Burton L. Mack goes as far as to say, in Who Wrote the New Testament, page 152, it would not have made sense before the war had run its course and the tragic fate of the city was known. The prophecies in Mark chapter 13 are clearly of the First Roman-Jewish War and of the civil war that raged concurrently within the Jerusalem walls, but these prophecies were written with after the event. Had Jesus prophesied the destruction of the temple, he would have been correct, but this prophecy was followed by another prophecy that he would return on clouds of glory within the lifetimes of those to whom he spoke, an event that did not occur as prophesied. Since it is not possible for Jesus to make a prophecy that does not come true, these were not his prophecies and were actually written at the end of the War.
Most scholars now believe that Matthew and Luke were substantially derived from Mark's Gospel. In fact, John Dominic Crossan, in The Birth of Christianity, page 110-111, speaks of a massive consensus among scholars in favour of Markan priority. On this information we can say that the Gospels of Matthew and Luke, being written some time later than Mark, were certainly written after 70 CE. Acts of the Apostles was written some time after Luke, although it does not mention the destruction of the temple nor the Roman-Jewish War of 66-70.
John's Gospel is generally dated 80-110 CE, although critical scholars, who see it as influenced by Luke's Gospel, would place in towards the later end of this range. The three Johannine epistles were written shortly after the Gospel.
Thirteen epistles have been attributed to the apostle Paul: Romans, 1 Corinthians, 2 Corinthians, Philemon, Galatians, Philippians and 1 Thessalonians, but five of these are regarded as disputed. Paul's genuine epistles were, of course, written before 70 CE. Known to critical scholars as pseudo-Pauline epistles, Ephesians, Colossians, 2 Thessalonians, 1 Timothy, 2 Timothy and Titus are thought to have been written between 70 Ce and the first half of the second century.
Critical scholars place the First Epistle of Peter no earlier than 80 CE, and the Epistle of Jude and the Second Epistle of Peter in the first half of the second century. Jude, perhaps carelessly, self-identifies as written long after the apostolic era, while scholars have noted that 2 Peter uses material from the earlier Epistle of Jude.
The Book of Revelation is almost universally accepted as having been written after 70 CE.
There are several scholars that have noticed the significance that no NT books record the destruction of the temple in 70 a.d. The only time it's mentioned is in prophecy. As John A.T. Robinson notes in his book Redating the New Testament:
"Explanations for this silence have of course been attempted. Yet the simplest explanation of all, that perhaps ... there is extremely little in the New Testament later than AD 70 [Moule, op. cit., 121.] and that its events are not mentioned because they had not yet occurred, seems to me to demand more attention than it has received in critical circles."
If you click on the book title above, it will take you to a lengthy excerpt of the first two chapters. Robinson quotes many other scholars and provides both external and internal evidence that most of the books were written prior to 70 a.d.
The debate really comes down to whether one believes that Yeshua predicted the destruction of the temple, or if these predictions were inserted ex eventu. Robinson makes a compelling argument that Yeshua did in fact predict it's destruction.
One of the biggest things I find significant is the ending of Acts:
"And Paul dwelt two whole years in his own hired house, and received all that came in unto him,
Preaching the kingdom of God, and teaching those things which concern the Lord Jesus Christ, with all confidence, no man forbidding him." Acts 28:30
All the author says is that Paul dwelled in his own hired house for two years. He doesn't say anything about what happened after those two years. Acts gives a very detailed story of the events after Yeshua's resurrection, and says "we" several times throughout (as though the author is remembering specific events that he experience himself), so if it was written after 70 a.d. it is amazing that the deaths of none of the apostles (except Jacob son of Zebedee) and the destruction of the temple are not even hinted at.
So my answer is that none of the NT books were written after the destruction of the temple (except for 2 Peter), and that the only time it's mentioned is when Yeshua prophesied it.
Too much has been made of what appears to what seems odd to many Christians--that no New Testament books refer to the destruction of the Temple. Why? Because, although arguments pro and con regarding whether the New Testament books are Jewish books, I think that from the earliest Christian books on (Paul's letters), it is apparent that they diverge from Judaism too much to be considered Jewish writings and that Christianity is beyond the pale of acceptable forms of Judaism.
Christianity has many serious disagreements and change of primary emphasis from Judaism. While many Christians say that the life, death, and resurrection of Jesus Christ was the most critical event for Jews in the first century, for Jews it was the destruction of the Temple. Christian writings had already made it clear that Temple sacrifices had been superseded by the sacrifice of Christ, the Lamb of God. Jews did not believe that all were fallen or that there even was a story of the Fall in Genesis 2-3 (read it literally and you will see that there isn't). Jews did not believe that anything more than the paths to redemption laid out in the Tanakh were necessary. If a Jew sinned, he could be redeemed by repentance, prayer, and maybe sacrifice. Repentance involved behavior--behaving with more righteousness, justice, and mercy. Christians proposed that what was now necessary was something, a belief, that was totally new: belief in the salvific value of Jesus' suffering, crucifixion and resurrection. To Jews that was ridiculous: you don't get in good standing with God by believing something but through righteousness. For first century Jews, the messiah would be known by being a figure of grandeur and power who would vanquish Israel's enemies, restore the nation Israel, and help God usher in his Kingdom. Christianity completely changed the meaning of "messiah" ("christ") and then blamed Jews for failing to recognize their messiah.
All of this is to say that Christians were not affected much by the destruction of the Temple whereas it changed everything for the Jews, forcing them in the next two centuries to become a People of the Book rather than of the sacrificial cult of the Temple. So why would Christian writers mention it? That they don't is much less odd than many make it out to be. It cannot serve well as paths of research useful for dating New Testament writings.
A common sense approach to the dating of the Gospels is that since Jesus clearly predicts the destruction of the Temple (Mark 13, Mathew 24 and Luke 21), the writers of the Gospels must have been writing after its destruction OR the Gospels were edited after its destruction.
The argument that they would surely have pointed to the actual destruction as evidence of the prophetic accuracy of Jesus is a weak one. Since the writers were writing in world where everyone knew about the fall of Jerusalem at the hands of the Roman Empire and consequent destruction of the Temple there would be no need to make that actual point.
All three synoptic Gospels use the same words - 'not one stone left upon another' - which suggests that all three of the Gospels are based on a common text written after the destruction of the Temple (AD 70). Many scholars suggest that this text is what makes up the core of Mark's gospel.