"I, even I, am the one who wipes out your transgressions for My own sake, And I will not remember your sins." Isaiah 43:25
"They will not teach again, each man his neighbor and each man his brother, saying, 'Know the LORD,' for they will all know Me, from the least of them to the greatest of them," declares the LORD, "for I will forgive their iniquity, and their sin I will remember no more." Jeremiah 31:34
"For I will be merciful to their iniquities, and I will remember their sins no more." Hebrews 8:12
"And their sins and their lawless deeds I will remember no more." Hebrews 10:17
According to these four verses, are we to assume that once we have been justified and saved by the grace of God through faith in the person and work of Christ Jesus, that God forgets our sins? No. That is not what these verses say. They do not say that God will forget our sins; they say that He will remember them no more. What this means is that God chooses not to remember them and hold them against us. In His mind, it is as if they never happened at all. But He does not forget them. God is omnipotent. He knows everything. He could not forget anything. Forgetting something and choosing not to remember something are two different things. In forgetting, it is a lapse of the memory. In choosing not to remember, while you will never forget, it is an act of the will.
Not only does God choose not to remember our sins, but we can do the same thing. When others sin against us, it is impossible for us to forget. The old cliché "Forgive and forget" is simply sheer nonsense. We are capable of forgiving trespasses done against us, as we are called and commanded to do (Matt. 6:12; Luke 11:4), but we will never be capable of forgetting. However, we can choose to remember those transgressions no more. We all know how our minds like to drag up past trespasses done against us so that we can cast them back in the teeth of others, reminding them of their failures and shortcomings in the past. We also know that doing so is not Christian behaviour and is contrary to how Christ Jesus wants us to show love toward one another (John 13:34-35; 15:12, 17; Rom. 12:10; 13:8; et al.).
What do I mean when I say that we, too, can choose to remember someone's sins against us no more? Let us say that your spouse has done something again to hurt you, whether intentionally or unintentionally. Automatically, your mind has already dug up all the previous times your spouse has hurt you. What I mean by saying that we, too, can choose to remember someone's sins against us no more, is that when our mind does this, we can choose to ignore those thoughts and put them out of our minds; we can dismiss them. If we are thinking on loftier things, such as Philippians 4:8 commands us to do, doing so will become easier. When your mind dredges up past trespasses, you can look at it as an opportunity to extend further forgiveness to that past sin and push it out of your mind. While you will never forget it, when it comes up, you can choose not to remember it. You do not have to grant it an audience to your thought process and dwell on it.
I am not saying that this will be easy, but like all Christian spiritual growth, it will take much effort and practice. Remember, we are to be emulating Christ Jesus. As Jesus forgives, we are to forgive. As Jesus chooses not to remember, we are to choose not to remember. If we desire Christ-likeness, we need to strive to be like Jesus.