Question: "Does the Bible instruct us to forgive and forget?"
Answer: The phrase "forgive and forget" is not found in the Bible. However, there are numerous scriptures commanding us to “forgive one another” (Matthew 6:14; Ephesians 4:32). A Christian who does not forgive can reap bitterness and the loss of eternal rewards (Hebrews 12:14-15; 2 John 1:8). Forgiveness is a decision of the will. Since God commands us to forgive, we must make a conscious choice to forgive. This frees the forgiving one from the past. The offender may not desire forgiveness and may not change (Matthew 5:44). Ideally, the offender will seek reconciliation, but if not, the one wronged should still make known his decision to forgive.
In one sense, it is impossible to truly forget sins that have been committed against us. We cannot selectively "delete" events from our memory. The Bible states that God does not "remember" our wickedness (Hebrews 8:12). God is all-knowing. God knows that we have “sinned and fall short of the glory of God” (Romans 3:23). However, having forgiven us, He treats us as if the sin had not occurred. If we belong to Him through faith in Christ, God does not hold our sins against us. In that sense we must "forgive and forget." If we forgive someone, we must act as if that sin had never occurred. We remember the sin, but we live as if we did not remember it. Ephesians 4:32 tells us, "Be kind and compassionate to one another, forgiving each other, just as in Christ God forgave you."