One answer from Jesus:
“But be on your guard. For they will deliver you over to councils, and you will be beaten in synagogues, and you will stand before governors and kings for my sake, to bear witness before them. And the gospel must first be proclaimed to all nations. And when they bring you to trial and deliver you over, do not be anxious beforehand what you are to say, but say whatever is given you in that hour, for it is not you who speak, but the Holy Spirit. And brother will deliver brother over to death, and the father his child, and children will rise against parents and have them put to death. And you will be hated by all for my name's sake. But the one who endures to the end will be saved.—Mark 13:9-13 (ESV)
To paraphrase: When you are accused of something, the Holy Spirit will guide how you should respond.
And another from Paul:
But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, self-control; against such things there is no law. And those who belong to Christ Jesus have crucified the flesh with its passions and desires.—Galatians 5:22-24 (ESV)
Which I take to mean: Do things the Spirit prompts and you won't be rightly accused of wrong-doing.
They don't have radically different answers and even agree that everything will turn out alright in the spiritual realm if not in the earthly one.
Both responded to false accusations in surprisingly pragmatic ways. Jesus defended Himself against accusations and attempts to murder Him while it was not yet the right time, but when it came time to fulfill His mission, He went quietly. Paul also seems to struggle in Philippians about whether he should fight the charges against him and certainly did stand up for himself over and over according to Acts. But when he went to Jerusalem the final time and was taken to Rome in chains, he did not shy away from doing what he felt called to do.
In my daily Bible reading today, I heard an answer from Peter:
Servants, be subject to your masters with all respect, not only to the good and gentle but also to the unjust. For this is a gracious thing, when, mindful of God, one endures sorrows while suffering unjustly. For what credit is it if, when you sin and are beaten for it, you endure? But if when you do good and suffer for it you endure, this is a gracious thing in the sight of God. For to this you have been called, because Christ also suffered for you, leaving you an example, so that you might follow in his steps. He committed no sin, neither was deceit found in his mouth. When he was reviled, he did not revile in return; when he suffered, he did not threaten, but continued entrusting himself to him who judges justly. He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed. For you were straying like sheep, but have now returned to the Shepherd and Overseer of your souls.—1st Peter 2:18-25 (ESV)
The entire letter of 1st Peter emphasizes that this sort of submission is expect of all Christians, whether slaves, servants or free people. (See especially 1st Peter 4:12-19.) However, Peter makes clear that we shouldn't blindly trust people who are in authority when he writes his second letter:
But false prophets also arose among the people, just as there will be false teachers among you, who will secretly bring in destructive heresies, even denying the Master who bought them, bringing upon themselves swift destruction. And many will follow their sensuality, and because of them the way of truth will be blasphemed. And in their greed they will exploit you with false words. Their condemnation from long ago is not idle, and their destruction is not asleep.—2nd Peter 2:1-3 (ESV)
From personal experience (and this is probably off-topic, but I'm compelled to mention it anyway), people sometimes get trapped in unhealthy situations: bosses who exploit them, spouses who abuse them, friends who lead them astray, etc. Even some pastors and other church leaders (who may be as sinful as the rest of us) sometimes use their authority to mislead (in every sense) their own flocks. I think the Christian response is to endure for a time (following Jesus, Paul and Peter), seek reconciliation, and, if the abusive relationship continues, seek guidance about how to step away.
My wife worked for a Christian non-profit for a few years and her immediate boss exploited her and her co-workers. She sought reconciliation over and over following the steps outlined by Our Lord in Matthew 18, and yet the unhealthy relationship continued. Finally on the the advice of several of our pastors, she quite the job and pursued another career.
We are sometimes too quick to abandon difficult positions, but its equal folly to hold onto a situation beyond what is good for our souls.
The right answer seems to be dependent on circumstances and the leading of the Holy Spirit. As always, seek wise counsel.