Contradiction with Matthew 22:15-22
This parallels the account in Luke but uses different language regarding their amazement.
Matthew 22:15-22: And they sent out unto him their disciples with the Herodians, saying, Master, we know that thou art true, and teachest the way of God in truth, neither carest thou for any [man]: for thou regardest not the person of men.
Contradiction with Mark 12:13-17
This is another parallel account, but it describes how the Pharisees and Herodians aimed to trap Jesus in a different manner.
Mark 12:13-17: And they send unto him certain of the Pharisees and of the Herodians, to catch him in [his] words.
Contradiction with Romans 12:18
This promotes living peaceably with all, which could be seen as contradicting the implications of Jesus' clever avoidance of conflict in Luke 20:26.
Romans 12:18: If it be possible, as much as lieth in you, live peaceably with all men.
Paradox #1
The possible contradiction or conflict could be the tension between the obligation to earthly authorities and the commitment to divine authority. People might struggle with understanding if their duties to government align with their religious beliefs, especially when the two seem to be in conflict.