What does Luke 20:21 mean?

The Jewish religious leaders are tempting Jesus to declare authority over the Romans. The lawyer-teachers and priests are jealous of the crowds who are following Jesus because He raised Lazarus from the dead (John 12:9–11). They seek a way to discredit Jesus' authority without angering the common people. If Jesus overtly broke Roman law, their enforcers would do the dirty work of Jesus' enemies (Luke 20:19–20).

To that end, the scribes and chief priests have planted spies in the crowd listening to Jesus' teaching. At the opportune time, a spy asks Jesus a seemingly innocuous question. In that culture, rabbis and students discuss hypothetical situations all the time. The difference here is that if Jesus answers wrong, He will be encouraging the crowd to break the Roman law.

The questioner is likely a follower of the Pharisees or Herodians. The flattering introduction is meant to make the man sound like one who respects Jesus. The assumption is that this will soften Jesus up so He won't notice the ramifications of the question. The question is, "Is it lawful for us to give tribute to Caesar, or not?" (Luke 20:22).

The assertion that Jesus shows "no partiality" is part of the trap. Based on recent history, Jesus is willing to condemn the Herodians, the Pharisees, and the Sadducees. The questioners hope that their question will pressure Jesus to antagonize Rome this time.

"Tribute to Caesar" just means the poll tax or census tax. It doesn't refer to any requirement to worship Caesar, which Jews of that era were not required to do. The underlying question is whether Jesus agrees that Jews should submit to Rome—which is what the Herodians and Sadducees would say. Alternatively, He might suggest the Jews refuse to pay the tax in a show of Jewish independence—which is what the Pharisees and Zealots would prefer.

Jesus' enemies aren't interested in an actual solution. They simply want Jesus to tell the people to revolt against Caesar, or to cave to the nation's hated oppressors.