Matthew 6:12 "And forgive us our debts, as we also have forgiven our debtors."
We pray that God forgives us of our debts(sins) AS we also forgive our debtors(those who sin against us). The "as" indicates in a similar manner. What is one thing that is necessary for God to forgive us of our sins? Genuine repentance. It's the same with those who sin against us. We'll see this made clear by what Jesus says at Matthew 18:21-34;
Matthew 18:21-34 “Then Peter came up and said to Him, “Lord, how many times shall my brother sin against me and I still forgive him? Up to seven times?” 22 Jesus said to him, “I do not say to you, up to seven times, but up to seventy-seven times. 23 “For this reason the kingdom of heaven is like a king who wanted to settle accounts with his slaves. 24 And when he had begun to settle them, one who owed him ten thousand talents was brought to him. 25 But since he did not have the means to repay, his master commanded that he be sold, along with his wife and children and all that he had, and repayment be made. 26 So the slave fell to the ground and prostrated himself before him, saying, ‘Have patience with me and I will repay you everything.’ 27 And the master of that slave felt compassion, and he released him and forgave him the debt. 28 But that slave went out and found one of his fellow slaves who owed him a hundred denarii; and he seized him and began to choke him, saying, ‘Pay back what you owe!’ 29 So his fellow slave fell to the ground and began to plead with him, saying, ‘Have patience with me and I will repay you.’ 30 But he was unwilling, and went and threw him in prison until he would pay back what was owed. 31 So when his fellow slaves saw what had happened, they were deeply grieved and came and reported to their master all that had happened. 32 Then summoning him, his master said to him, ‘You wicked slave, I forgave you all that debt because you pleaded with me. 33 Should you not also have had mercy on your fellow slave, in the same way that I had mercy on you?’ 34 And his master, moved with anger, handed him over to the torturers until he would repay all that was owed him. 35 My heavenly Father will also do the same to you, if each of you does not forgive his brother from your heart.”
What did Jesus mean when he said, for this reason the kingdom of heaven is like a king who wanted to settle accounts with his slaves? What's the reason Jesus is talking about? The necessity of forgiveness where there is repentance. This is made plain in the parable Jesus told. What led the master to have compassion and forgive the slave? The slave "fell to the ground and prostrated himself" before the master, and begged for the patience of his master. The king himself said in verse 32 that he forgave the slave of all his debt because he pleaded with him.
The master could discern that the slave was genuinely repentant by what he did, i.e. fall to the ground, prostrate himself, and beg for mercy. The genuine repentance on the part of the slave enabled the master to forgive the debt entirely; obviously, if the slave did not show any signs of repentance(i.e. prostrating, begging for patience), the master would have never forgiven him of his debts.
There are three essential points that Jesus is trying to make with this parable;
(1) In order for there to be forgiveness, there first has to be repentance. That is a stipulation. This applies to both the forgiveness of humans(remember, it was Peter who incited Jesus to tell this parable by his question of how many times he should forgive his brother) and the forgiveness of God(the king in the parable represents God, and the king forgave his slave because he pleaded with him. Thus, God forgives us because we plead with him for forgiveness, i.e. are repentant).
(2) Forgiveness on the part of God is necessitated when there is repentance on the part of the one who sins(i.e. for this reason the kingdom of heaven is like a king who wanted to settle accounts with his slaves...).
(3) Because God always forgives us when we are genuinely repentant, we must forgive others when they are repentant, otherwise, God will stop forgiving us, just as the king stopped forgiving his slave(i.e. my heavenly Father will also do the same to you, if each of you does not forgive his brother...).
Luke 17:3-4 makes point 1 unequivocal;
Luke 17:3-4 “Be on your guard! If your brother sins, rebuke him; and if he repents, forgive him. 4 And if he sins against you seven times a day, and returns to you seven times, saying, ‘I repent,’ you shall forgive him.””
Notice Jesus does not say, "if your brother sins, just forgive him no matter what"; he says, if your brothers sins AND repents, you must forgive him. Thus, if our brother repents of his sin against us, we are required to forgive him. It's as unambiguous as that. We are not required to forgive everything everyone does to us; we are only required to forgive the genuinely repentant. If we forgive someone who is unrepentant, and, being unrepentant, continues to sin, are we not encouraging them to continue in their ways? If God chooses to just let go of everyone's sin no matter what, then there would be no need for atonement, i.e. no need for what Jesus did on the cross. Everyone would be saved! And we could do whatever we want, live whatever kind of life, and never have to worry about the wrath of God being on us. Remember, we forgive our debtors AS God forgives us, i.e. in the same manner. If God does not forgive unrepentant ones, neither should we. It's as simple as that.
Now, knowing everything we know, let's analyze each of the situations given in the OP's question, and see how they fit within the framework already laid out.
- Person A sins against person B. B knows what they have done but A does not ask for forgiveness.
If person A does not ask for forgiveness, it is because he does not want it. If he did, all he'd have to do is ask; but he didn't, therefore he doesn't want it. What does this imply? That person A is unrepentant. Remember, you cannot forgive if there is no repentance. You cannot give someone something that they do not want, i.e. one cannot genuinely accept what they do not want. In order for person A to accept forgiveness, he'd have to want it, and it's clear that he doesn't, given the fact that he never requested it. Thus, in this circumstance, person B cannot forgive person A.
- A sins against B. B does not know what A has done and A does not ask for forgiveness.
If person B is not aware of the sin of person A, how is person B supposed to forgive person A of his sin? You can't forgive someone for something if you are completely unaware of what that something is! Thus, in this circumstance, person B cannot forgive person A.
- A sins against B. A asks for forgiveness but B has proof of their insincerity.
Well, then person B has to juggle between a type 1 and type 2 error. What's that? A type 1 error, also called a false positive, occurs when one rejects the null hypothesis when it is true, e.g. the null hypothesis of the given situation is [person A is repentant]; if the hypothesis is true, but person B rejects it and chooses to not forgive person A, he has made a type 1 error. A type 2 error, also called a false negative, occurs when one fails to reject(i.e. accepts) the null hypothesis when it is false, e.g. the null hypothesis of the given situation is [person A is repentant]; if the hypothesis is false, but person B accepts it and chooses to forgive person A, he has made a type 2 error.
In the given situation, there is no way to know with absolute certainty whether the hypothesis[person A is repentant] is true or not... but then again, there is never any way to know with absolute certainty whether someone is repentant or not. Human beings do not have the ability to search the hearts or examine the minds of men; only God does.
Jeremiah 17:10 "I the LORD search the heart and examine the mind, to reward each person according to their conduct, according to what their deeds deserve."
Remember, repentance isn't just saying your sorry; it's a genuine change in the condition of a person's heart. This will be made manifest in the things someone says and does. But it's not like it's impossible for a person to act in a way that contradicts what they know or believe(that is called cognitive dissonance); nor is it impossible for one to say they feel a certain way, when in actuality they do not(that is called a lie). The bottom line is, you have to infer to the best of your abilities whether or not someone is genuinely repentant. You will never be able to know with absolute certainty what one genuinely thinks or believes; only God can do so. Thus, all we can do is try our best, using all the information available to us, to ascertain the true condition of one's heart, all the while hoping to avoid making either a type 1 or type 2 error, always keeping in mind that we are imperfect and highly limited and thus prone to error, never forgetting that our Heavenly Father understands our imperfections and limitations and does not hold them against us... Indeed, life is NOT simple, not by any means.
Now, if there really is proof that person A is unrepentant, then (depending on how incriminating the proof is) it's most probable that the null hypothesis[person A is repentant] is false, and should thus be rejected. Once again, there's no way to know with 100% certainty, and that's alright; we must embrace the fact that we are imperfect, highly limited beings. But if the evidence highly substantiates the claim that person A is unrepentant, it would be inadvisable to reject it. Thus, in this circumstance, person B should not forgive person A.
- A sins against B. But A dies before B realizes what has happened.
If A dies before B realizes what has happened, there's really no way at all for person B to even guess if person A was repentant. Certainly, person A cannot repent while dead(or perhaps he can, depending on whether or not you believe the dead are conscious. But even if he does repent while dead, person B would never know). The only way person B can make a judgment on whether or not person A was repentant is by using what he knows about what person A did while still alive, i.e. whether or not he was ever remorseful for his actions, or ever pleaded for forgiveness. But as per what is given, none of that happened while person A was alive, as person B was unaware of person A's actions while he was still alive(if person A had shown remorse or asked for forgiveness, surely person B would have known about the sin person A committed against him; but he doesn't, thus person A could not have shown any remorse or have asked for forgiveness). Thus, in this circumstance, person B is incapable of forgiving person A.
Hope this helps! Have a great day. :)