John Gill Commentary Hebrews 8:12

"For I will be merciful to their iniquities, And their sins will I remember no more." — Hebrews 8:12 (ASV)

For I will be merciful to their unrighteousness
That is, sin; for all unrighteousness is sin, being contrary to the justice of God, and his righteous law: and the phrase is expressive of God's forgiveness of it, which is a very considerable article of the covenant of grace; mercy is the spring and original of pardon; it is what God delights in, and therefore he pardons freely; it is large and abundant, and hence he pardons fully; and this lays a foundation for hope in sensible sinners.

And the way and means, in and by which God pardons, is the propitiatory sacrifice of his Son; and the word here rendered "merciful", signifies "propitious"; God pardons none but those to whom he is pacified, or rendered propitious by Christ; there is no mercy, nor pardon, but through him; he pardons on the foot of reconciliation and satisfaction for sin by Christ.

So that forgiveness of sin is an act of justice, as well as of mercy; or it is an act of mercy streaming through the blood and sacrifice of Christ.

And their sins and their iniquities will I remember no more ;
by which are meant all kind of sin, original and actual; sins before and after conversion; every sin but that against the Holy Ghost, and that God's covenant people are never guilty of; these God remembers no more; he casts them behind his back, and into the depths of the sea, so that when they are sought for, they shall not be found; God will never charge them with them, or punish them for them:

This is another phrase to express the forgiveness of sins, and distinguishes the new covenant from the old one, or the former dispensation; in which, though there were many typical sacrifices, and a typical removal of sin, yet there was a remembrance of it every year.