Yes, God forgets our sins in the way the Bible describes God’s act of “forgetting.”
But, no, God doesn’t forget our sins in the way we humans forget things.
So when we say God “forgets our sins” (and I think it is okay to say that, since the Bible uses that language), we must be sure to qualify what we mean. If such language is flippantly used, it may result in severe misunderstanding.
The most helpful text in this discussion is Jeremiah 31:34, which includes this parallelism at the end:
“For I will forgive their wickedness
and will remember their sins no more”
In poetic and prophetic texts, synonymous parallelism is a common literary device that usually results in two mutually-interpreting phrases.
In this case, the Lord is not saying he will forgive wickedness and then, secondly, not remember their sins. This verse is not describing two different divine actions.
Instead, the word “forgive” is synonymous with “remember…no more.” Seeing this synonymous parallelism is reinforced by the objects of the two verbs: “their wickedness” is synonymous with “their sins.”
God doesn’t forget our sins in the way that we might forget what we received as birthday presents two years ago. God also doesn’t choose to limit his omniscience in order to see us as “righteous in Christ.” Such understandings about human forgetfulness unfortunately take the Bible’s anthropomorphic language about God too literally.
The Bible’s description of God “forgetting our sins” is simply another way of saying God “forgives our sins.”
And, by the way, it wouldn’t be good if God actually forgot (in the way that we forget) our sins, because that would compromise his omniscience. Consider this: if God actually forgets our sins, while we remember them pretty clearly, then we would be affirming that we know something God doesn’t know! However, the Bible doesn’t even entertain the possibility that man has any knowledge that God himself may lack.
So should Christians say “God forgets our sins”? Well, there’s no biblical reason not to, and there’s even biblical precedent for it (Jer 31:34). Just don’t use flippantly unqualified biblical language that could be wrongly interpreted as compromising another biblical truth (namely, God’s omniscience).
As Christians, we celebrate the awesome reality that God knows everything–including every one of our sins. But as believers, we also celebrate the fact that God no longer counts our iniquities against us (Rom 8:1).