What do you do if you make a wrong point in a religious debate or end up misquoting something or committing a logical fallacy?

You humbly accept correction.

Admit it.

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I would continue to make my point "virtually" according to Zechariah 1:15-16. Selah <Can More Than a Khanqueror Uncle Travus Tsuper kuiper Juan Amongst a Trillion DuMore Good by Any and ALL Holy Means Necessary in The Almighty Name of The LORD YaHaVaH Who Was Is and Will Be The Only True God and None Other Than our Great God The Righteous Tsalvation of Justice Anointed KING of kings LORD of lords GOD of god The Tsupreme and The Only Potentate Tzedekelizedek HaMashiach a High Priest Forever and Ever and Evermore According to the Order of Melchizedek Amen Hallelujah i Believe>

Everything after your "OR" is what you should do.  This should be the case in all debates, regardless of topic.

Admit the error so people won't get deceived by incorrect information, apologize, and continue making your point.

So, are you saying that the debate is religious?

Well!

On true faith in God, religious people have many things wrong, it is not so, with True Christians Faithful to God, their truth is based on sound doctrine, imparted by our Lord Jesus Christ.

If you are mistaken, and it can be adequately demonstrated that you are in error then you acknowledge you are wrong and you change your mind (you do not just continue to make your provably false point).  Every time you can be shown evidence that you are wrong, you must tear apart and rethink the very foundations that get you to your current belief from the ground up.  Then you must adjust your beliefs and the foundations of your beliefs to most adequately describe reality.  If you truly care about knowing what is true then facts are more important than what you want to be true or what you think is true.

In an argument we are arguing in order to establish true propositions.  If your propositions are not true then concede the argument.  It may be that your interlocuter is not correct either but you must admit that your position is wrong and you will have to rethink that point in question.

Could you give an example? I mean, if that happens it sounds like you need better command of the subject. Lastly, I probably wouldn't concede the point regardless but end up standing my ground.

Religious debates rarely result in much more than breaking down because of differing dogmas.  What we use to define large words with deep and nuanced meaning is our dogma.  If we do not share the same agreed definition to these types of words then it is like speaking a different language.

When we live a repentant life we are routinely seeking help with our faults from God.  This attitude is reflected the most in times of conflict or crisis.  The repentant soul will not seek to get better at your expense.  Their goal is always to see you as an equal.

We have the right to ask anyone making claims about God to document those claims in God's Word.  If they are unable to do that then they are fools or liars.  We are to be open to people seeking the reason for the hope inside our hearts.  We are to avoid playing games with fools or entertaining liars.  Context matters very much in how we treat people who "make mistakes" or misrepresent facts pertaining to religion.

The answer depends on the context.  The people involved and their motives.  The details of the mistake.  All of these change how we should react.

Take the atheist position and you can't go wrong.  Religions are built on fallacies.