Luke, like Matthew and Mark, contains an extensive discourse about the destruction of the Jewish temple. You can read his version in Luke 21:5-36. The parallels are Matthew 24 and Mark 13.
Now here is the way Luke’s Gospel ends:
50 Then he [Jesus] led them out as far as Bethany, and lifting up his hands he blessed them. 51 While he blessed them, he parted from them and was carried up into heaven.
52 And they worshiped him, and returned to Jerusalem with great joy, 53 and were continually in the temple blessing God [Luke 24:50-53].
While it’s true, historically, that the disciples continued to worship in the temple for some time, Luke didn’t have to end his Gospel with a mention of the temple (the other Evangelists didn’t)–and it’s such a positive mention in view of Jesus’ prophecy of its destruction just a few chapters earlier.
If that prophecy had already been fulfilled when Luke wrote his Gospel, it would be a little odd for Luke to end with the image of the disciples happily worshipping in the temple.
After the temple was destroyed by the Romans in A.D. 70, it was forcefully impressed on Christian consciousness that the Crucifixion and Resurrection of Jesus spelt the end of the temple and its system of worship. That didn’t really sink in until the temple was destroyed, as Acts records the disciples (including Paul) continuing to worship in the temple for decades (until the mid-A.D. 50s, when Paul gets arrested and the story shifts away from Jerusalem for the last time).
Luke’s ending seems more like the way the temple and its services would be portrayed before it was destroyed–when Christians still frequented it and regarded it as the House of God.
This is not a decisive argument for when Luke was written, but it is “a straw in the wind” that points to it being composed before A.D. 70.
Fortunately, we have other, stronger reasons to suppose that. (It was most likely written around A.D. 59, since Acts stops suddenly in A.D. 60.)
