Bible Answer

If our sins are not “remembered” by God, how can He judge believers?

In your teaching from Hebrews, you say that a believer's failure will be judged by God. But the Bible says that God will "remember our sins no more" and that our sins are separated from us as far as the East is from the West? How can the Gospel be Good News if God is remembering our sins?

You are conflating two different judgments. The Christian is forgiven of sin by the blood of Christ, so our sins are not "remembered" by God in the sense that He will not hold us accountable for them with the penalty of the Second Death. This is the Great White Throne Judgement and believers will not face this judgment:

Rev. 20:6 Blessed and holy is the one who has a part in the first resurrection; over these the second death has no power, but they will be priests of God and of Christ and will reign with Him for a thousand years.

However, there is another judgment for the believer in which Jesus determines our eternal rewards:

2Cor. 5:9 Therefore we also have as our ambition, whether at home or absent, to be pleasing to Him
2Cor. 5:10 For we must all appear before the judgment seat of Christ, so that each one may be recompensed for his deeds in the body, according to what he has done, whether good or bad.

This is the Judgment Seat of Christ, which takes place at the resurrection (i.e., Rapture) of the Church. This judgment is for the basis of rewards, and as Paul says, it takes into account those things we've done, whether good or bad. A Christian who lives a life of disobedience will see eternal consequences in the form of a loss of eternal reward. At that time, all our works, good and bad, are in view for that evaluation, Paul says:

1Cor. 4:5 Therefore do not go on passing judgment before the time, but wait until the Lord comes who will both bring to light the things hidden in the darkness and disclose the motives of men’s hearts; and then each man’s praise will come to him from God.

Once again, this is not a judgment for determining salvation, for salvation is determined by faith alone and not by works. But our works are evaluated for eternal reward, and that is the concern of Hebrews 4. So the GOSPEL is very good news indeed for all who come to faith in Jesus Christ, and now that we are called by His name, we are to serve our Master obediently. Failure to serve the God Who saved you is not good as the writer of Hebrews says later:

Heb. 10:26  For if we go on sinning willfully after receiving the knowledge of the truth, there no longer remains a sacrifice for sins
Heb. 10:27 but a terrifying expectation of judgment and THE FURY OF A FIRE WHICH WILL CONSUME THE ADVERSARIES
Heb. 10:28 Anyone who has set aside the Law of Moses dies without mercy on the testimony of two or three witnesses
Heb. 10:29 How much severer punishment do you think he will deserve who has trampled under foot the Son of God, and has regarded as unclean the blood of the covenant by which he was sanctified, and has insulted the Spirit of grace
Heb. 10:30 For we know Him who said, “VENGEANCE IS MINE, I WILL REPAY.” And again, “THE LORD WILL JUDGE HIS PEOPLE.”
Heb. 10:31 It is a terrifying thing to fall into the hands of the living God.

That passage was written to believers (notice in v.31 the writer says God will judge "His" people). The modern church has done the believer a great disservice by downplaying or ignoring teaching on the reality of a judgment for the believer, which has allowed generations of believers to remain ignorant of their obligations to serve Christ. As Peter says, the believer will be the first to be judged:

1Pet. 1:17  If you address as Father the One who impartially judges according to each one’s work, conduct yourselves in fear during the time of your stay on earth
1Pet. 1:18 knowing that you were not redeemed with perishable things like silver or gold from your futile way of life inherited from your forefathers
1Pet. 1:19 but with precious blood, as of a lamb unblemished and spotless, the blood of Christ.
1Pet. 4:17 For it is time for judgment to begin with the household of God; and if it begins with us first, what will be the outcome for those who do not obey the gospel of God?

For more background on eternal rewards and the judgment believers will face, please read our article on Eternal Rewards.