Contradiction with Matthew 5:39
Acts 25:10 speaks about Paul's appeal to the authority of Caesar for justice, while Matthew 5:39 advises against resisting evil and instead suggests turning the other cheek.
Matthew 5:39: But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
Contradiction with Romans 13:1
Romans 13:1 encourages submission to governing authorities because they are instituted by God, whereas Acts 25:10 has Paul make an appeal to Caesar, using his rights rather than submitting to the local authority's judgment.
Romans 13:1: Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God. [ordained: or, ordered]
Contradiction with 1 Corinthians 6:7
Acts 25:10 shows Paul's willingness to use the legal system to defend himself, but 1 Corinthians 6:7 suggests it is better to suffer wrong or be defrauded than to take a dispute to court, which Paul actively avoids by appealing to Caesar.
1 Corinthians 6:7: Now therefore there is utterly a fault among you, because ye go to law one with another. Why do ye not rather take wrong? why do ye not rather [suffer yourselves to] be defrauded?