What are the arguments that the book of Acts was written after 70 AD?

The best place to start is with the history of the New Testament books. Nearly all New Testament commentators agree that Acts was written after Luke's Gospel and by the same anonymous author. So the next step is to establish whether Luke was written no earlier than 70 CE. I believe that this can be established by looking at two of Luke's sources and some archaeological evidence.

Mark
Adam Winn (The Purpose of Mark's Gospel, page 1) says there is a high level of consensus among New Testament interpreters in favour of the theory of Markan priority. Some excellent answers to another question, What are the arguments in favor of Markan priority? cover just some of the reasons for this conclusion. Here, I will touch on some of the evidence to show that Luke (in particular) has a literary dependence on Mark, but this can not be exhaustive in this format.

The 'Missing Block' is a substantial section of Mark, comprising a total of 74.5 verses from Mark 6:47 to Mark 8:27a, missing from Luke. This results in the curious conjunction found in Luke 9:18 "And it came to pass as he was alone praying, his disciples were with him: and he asked them ..." These clauses are more meaningful when found in Mark at the start and end of the material missing from Luke.

A technical proof is in the transmision of Markan intercalations, as explained by John Dominic Crossan in The Birth of Christianity, page 106. Simplifying a little, intercalations were a literary device in the form A1-B-A2, which adds emphasis to both event A and event B. Of the nine undisputed Markan intercalations, Matthew retains them five times and Luke does four times, otherwise copying the content but not the structure. It is clear from this that the device moves from Mark to Matthew and to Luke, but not the reverse.

If the priority of Mark is established, the next step is to determine when Mark's Gospel was written. Burton L. Mack says, in Who Wrote the New Testament, page 152, the account in Mark 13:1-12 would not have made sense before the First Roman-Jewish War had run its course and the tragic fate of the city was known - which places the authorship of Mark around 70 CE. This is the consensus of most critical scholars. Then, it would almost certainly have taken at least 10 or 20 years for the first New Testament gospel to circulate and come to the notice of Luke's author.

Josephus
Steve Mason,in Josephus and the New Testament, chapter 6, examines some striking parallels of structure, aim and vocabulary between Josephus and Luke-Acts. If that is evidence of copying from Josephus to Luke-Acts, then Acts could scarcely have been written before the very end of the first century.

Matthew Ryan Hauge (The Biblical Tour of Hell, page 55) says that two recent studies suggest 'Luke' advanced to the early stages of a rhetorical education, which means that he was trained in the art of copying and the complexities of rhetorical composition through mimesis. One way of identifying the existence of copying or imitation is to locate any mimetic flags that may exist. Greek students were taught to insert mimetic flags that could identify their sources, to avoid charges of plagiarism. If present, a mimetic flag will be unusual for that particular genre and context. Some scholars see Luke's introductions (Luke 1:1-3, Acts 1:1), with their dedications to Theophilus, as mimetic flags identifying Josephus as one of his sources. As the only New Testament books to have such a dedication, they are unusual for this genre as well as being unusual for such short works. By comparison, longer works, such of those of Josephus, often mentioned a patron. Josephus dedicated Jewish Antiquities, Life and Against Apion to Epaphroditus. The parallel is particularly evident in Acts which looks back to the first book, much as Josephus looked back to Jewish Antiquities, in Life and Against Apion. Theophilus was a common name that means 'Friend of God', which has a strong resemblance to Epaphroditus, a common name that means 'Touched by Aphrodite'. This may not prove that Luke copied from Josephus, but is strong evidence that this may be the case.

Richard Carrier has written a good summary of some evidence in Steve Mason's book mentioned earlier. I think their most compelling evidence that Acts must have copied from Josephus is found in its mention of some revolutionaries. Josephus mentioned that there were many "deceivers and imposters" who led the Jewish people into revolt, giving three examples: Judas the Galilean; Theudas, who led a group of revolutionaries (Antiquities 20:5:1); and an unnamed Egyptian prophet. If there were "many deceivers and impostors", Luke need not have chosen the same names unless he only knew the names from Josephus, yet these are the very three names found in Acts (Acts 5:36-37 and Acts 21:38). Even by itself, this strongly suggests that Josephus' works were Luke's only sources for that period. Josephus says that Judas the Galilean was first (around 6 CE), Theudas was next (circa 44-46 CE) and the Egyptian was third (52-59 CE) but for literary reasons, in Antiquities XX.5.1-2, Josephus mentions Theudas before Judas the Galilean, while making it clear that Judas lived a generation before Theudas. Acts tells of Theudas and Judas in the wrong chronological order, because Luke was following the order in which Josephus wrote of them, certain evidence of literary dependence:

Acts 5:36-37: For before these days rose up Theudas, boasting himself to be somebody; to whom a number of men, about four hundred, joined themselves: who was slain; and all, as many as obeyed him, were scattered, and brought to nought. After this man rose up Judas of Galilee in the days of the taxing ...

Gallius
Acts refers to the judgement seat of Gallius:

Acts 18:12 And when Gallio was the deputy of Achaia, the Jews made insurrection with one accord against Paul, and brought him to the judgment seat

The Westar Seminar on the Acts of the Apostles (Spring report 2007) says the Gallio inscription has long been thought to provide confirmation of the historicity of the story of Paul’s hearing before Gallio. The inscription provides independent evidence that Gallio was proconsul in Corinth and indicates that he was there in the early 50s. So, when archeologists discovered a bema (a platform or dais that was used for public speeches or proclamations) in Corinth, it was interpreted to be the very bema before which Paul was judged by Gallio as detailed in Acts.

The Seminar says that L. Michael White presented new evidence that disputes this longstanding conclusion, with a reassessment of the archaeological data, suggesting that the bema was constructed after the time when Paul was there. If there was no bema, then, the Acts story can not be historical. Instead it would seem that Luke observed the existence of a bema in Corinth in his own day, in the early second century, added data he had gathered about the governorship of Gallio, and out of this created the story we have in Acts.