Sacrifice Can't Remove Sin
Today's passage really gets to the heart of the OT system, the one that the writer of Hebrews is urging the people to abandon. This passage said that sacrifice couldn't take away sin, so what did the sacrifices do?
"The law’s sacrificial system, rather than delivering worshipers from guilt, actually has the effect of reminding them of their sinfulness and, thus, their constant separation from God (v. 3). Why is this the case under the older covenant? Because the sacrifices of that system—“the blood of bulls and goats”—do not have the ability to remove sins (v. 4). The author already has made the point that the old sacrifices could “sanctify” or “purify” (9:13, 23) people, but here, significantly, he uses “take away” or “remove” (aphaireo), a word used with reference to sin only one other place in the New Testament. At Romans 11:26–27 Paul quotes the prophet Isaiah saying, “The deliverer will come from Zion; he will turn godlessness away from Jacob. And this is my covenant with them when I take away [a form of aphaireo] their sins.”
In Hebrews 10:4, as in the Romans passage, the idea of “removing” sin speaks of the burden sin placed on the worshiper’s conscience being lifted in a decisive, perpetually effective cleansing, which establishes one’s status before God. This is what the old covenant sacrifices were unable to do, which is why sin remained a separator, a perennial, detrimental force disallowing a permanently right relationship between God and his people."  (emphasis mine)
Just a few more thoughts from me on this topic: check out the next group of verses. It says "Sacrifices and offerings You have not desired..." If God doesn't desire these things, why were the people to offer them? The OT has a number of statements that are similar to this one, which seems weird because God tells them to sacrifice, right? I believe that those who truly understood the OT sacrificial system understood this fact. Of course there were plenty of people that just sacrificed because it was the thing to do - they had no real understanding of it. That's not what God desires. "For I desire steadfast love and not sacrifice, the knowledge of God rather than burnt offerings." (Hosea 6:6, ESV) I think that those that understood the system knew that it was not actually the SACRIFICE that cleansed them, but GOD. God said "If you want to be cleansed, do ______". Those that did it out of duty focused on the ______, while those that understood the truth realized that ultimately it is God and His promise that cleanses and removes sin. They understood that God doesn't want sacrifices, but for people to know Him. He wants love, not simply duty.
This is vitally important because it means that God has never truly changed the system of salvation. It was God and His promise that removed sin before, and it is God and His promise through the sacrifice of Jesus that removes sin now. Christ has come in full revelation, so these "shadows" of true forgiveness don't make sense anymore - they have become obsolete.
But don't think that this is just for the OT - it has a lot of application for today. Relationship with God and forgiveness of sin has nothing to do with our "sacrifices" (money, service, time, etc.) Just as before, God desires love and knowledge of Him, not these trifles. These "sacrifices" only make sense when we do them out of love and gratitude, not because we think we should or because we'll be "good" with God if we do.
Only Jesus makes us good with God, and that's what makes Him so much greater than these other things in Hebrews!