There is a lot of attention given to our memory. What used to be called senility and was left basically untreated is now called dementia and Alzheimer's and given much medical treatment and research. There are websites devoted to mental games to help keep our minds sharp. There are herbal and other medicines for it as well. We take pictures so we can remember. Some things are important to remember, but others are better forgotten. By forgotten I mean not dwelt on. We don't usually actually forget important or impactful moments or situations, but we can forget them in the sense we do not allow them control over us.
God does not forget. He remembers His covenant with His people (Deut 4:31, Is 44:21, Is 49:14-16, Lk 12:6-7). He remembers the good things we do (Mt 10:42, Heb 6:10). He remembers our prayers (1Sam 1:11, Ps 10:12, Ps 13:1, Ps 42:9, Ps 74:19 & 23, Ps 77:9, Rev 8:1-2). He remembers to judge those that forget Him (Ps 10:11, Jer 23:39, Lam 5:20, Hos 4:6, Amos 6:7). However, He chooses to forget the sins we have repented of (Ps 103:12, Is 38:17, Is 65:16, Heb 8:12).
We must never forget Him (Deut 32:18, Ps 9:17, Ps 50:22, Jer 2:32, Jer 23:27, Eze 22:12, Eze 23:35, Hos 8:4, Hos 13:6). We must never forget His covenant with us (Deut 4:9 & 23, Deut 6:12, Deut 8:11-19, Deut 26:13, 2Ki 17:38, Ps 44:17 & 20, Ps 78:7, Ps 119:16, 61, 83, 93, 109, 139, 141, 176, Prov 3:1, Prov 31:5, Is 17:10, Jer 3:21, Jer 13:25, Jer 18:15, Jer 23:27, Lam 2:6, Hos 4:6). We should remember His benefits (Ps 103:2), His city (Ps 137:5, Is 65:11, Jer 50:6), His wisdom (Prov 4:5), or His judgment (Ps 59:11). We should not forget to do good (Heb 13:16), or the exhortation of the word of God (Heb 12:5)
However, there are some things we do need to forget. We need to forget how to sin (Ps 45:10, Deut 12:3, Gal 2:18). We need to forgive others (Mt 6:12-15, Mt 18). They saying goes, "forgive and forget", but that is not what the Bible says to do. We are not commanded to forget in the sense of not remembering what happened. We can recall events but be free from their control. Joseph remembered what his brothers did to him, but he forgave them (Gen 42-50). We need to forgive ourselves. That does not mean we have to try to blot our mistakes out of our minds. In fact, it is good to remember for the purpose of learning from our mistakes. God even commanded this (Deut 9:7, Deut 31:21, Jer 44:9, 2Pet 1:9). Even better than learning from our mistakes is to learn from others' (1Cor10:1-13).