Short answer
It does, but the Law of Moses itself forgave sin without blood. The same verse says only that "almost all things", not all, were cleansed with blood under the Law. A man too poor to offer two birds could bring a little flour for certain sins, and "it shall be forgiven him". No animal died. The prophets make turning back to God (﷾) the way to forgiveness. Nineveh repented, no sacrifice is recorded, and God (﷾) spared it. Jesus twice quoted God's (﷾) words, "I will have mercy, and not sacrifice". In his own parable, a tax collector simply asked for mercy and went home forgiven. The Qur'an teaches the same. God (﷾) forgives the one who turns to Him, and no one carries another person's sin.
Full explanation
How Christians read Hebrews 9:22
Christians read this verse as a rule for all time: sin is forgiven only through shed blood. They add Leviticus 17:11, where God (﷾) says the blood makes atonement on the altar. From these they conclude that the animal sacrifices pointed ahead to the death of Jesus, and that forgiveness now comes through his blood.
“And almost all things are by the law purged with blood; and without shedding of blood is no remission.”
“For the life of the flesh is in the blood: and I have given it to you upon the altar to make an atonement for your souls: for it is the blood that maketh an atonement for the soul.”
The word "almost" and the flour offering
The writer of Hebrews himself says "almost all things" were cleansed with blood. He does not say every case was cleansed with blood, and the Law he describes has a clear case of forgiveness without it. For certain sins listed in Leviticus 5, a man could not afford a lamb or two birds. He brought a tenth of an ephah of fine flour instead. The priest burned a handful of it on the altar. The text says the sin "shall be forgiven him". Nothing was killed. So the Law itself does not make blood the only way.
“And almost all things are by the law purged with blood; and without shedding of blood is no remission.”
“But if he be not able to bring two turtledoves, or two young pigeons, then he that sinned shall bring for his offering the tenth part of an ephah of fine flour for a sin offering; he shall put no oil upon it, neither shall he put any frankincense thereon: for it is a sin offering. Then shall he bring it to the priest, and the priest shall take his handful of it, even a memorial thereof, and burn it on the altar, according to the offerings made by fire unto the Lord: it is a sin offering. And the priest shall make an atonement for him as touching his sin that he hath sinned in one of these, and it shall be forgiven him: and the remnant shall be the priest’s, as a meat offering.”
Repentance in the prophets
The prophets teach the same thing. Ezekiel, a Hebrew prophet of the sixth century BC, gives the way back to God (﷾) in plain words. When the wicked man turns from his sins and does what is right, his past sins "shall not be mentioned unto him". He saves his soul alive. There is no sacrifice in the passage and no one punished in his place. Nineveh was a city of non-Israelites, and the text records no sacrifice there. Its people fasted and turned from evil, and God (﷾) saw their works and spared them.
“But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. All his transgressions that he hath committed, they shall not be mentioned unto him: in his righteousness that he hath done he shall live.”
“Again, when the wicked man turneth away from his wickedness that he hath committed, and doeth that which is lawful and right, he shall save his soul alive.”
“And God saw their works, that they turned from their evil way; and God repented of the evil, that he had said that he would do unto them; and he did it not.”
"Mercy, and not sacrifice"
Jesus said the same. Hosea, a Hebrew prophet of the eighth century BC, records God (﷾) saying He desired mercy and not sacrifice. Jesus quoted these words twice in Matthew. In his parables, forgiveness comes without blood. The prodigal son's father runs to meet him and forgives him on the spot. The tax collector says only "God be merciful to me a sinner", and Jesus says he went home justified. None of these passages points ahead to a sacrifice.
“For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.”
“But go ye and learn what that meaneth, I will have mercy, and not sacrifice: for I am not come to call the righteous, but sinners to repentance.”
“But if ye had known what this meaneth, I will have mercy, and not sacrifice, ye would not have condemned the guiltless.”
“And he arose, and came to his father. But when he was yet a great way off, his father saw him, and had compassion, and ran, and fell on his neck, and kissed him.”
“And the publican, standing afar off, would not lift up so much as his eyes unto heaven, but smote upon his breast, saying, God be merciful to me a sinner. I tell you, this man went down to his house justified rather than the other: for every one that exalteth himself shall be abased; and he that humbleth himself shall be exalted.”
No one can carry another's sin
When a substitute was offered for sin, God (﷾) refused it. After the Israelites made and worshipped a golden calf, Moses asked God (﷾) to forgive his people, or else blot him out of His book. God (﷾) answered that He would blot out whoever had sinned against Him. The reason He gave was not that Moses had sinned. He gave a rule: the one who sinned answers for his own sin. No just court would punish an innocent man in place of the guilty, even a willing one. Ezekiel says the son shall not bear the iniquity of the father. The Qur'an says the same: no bearer of burdens can bear the burden of another. The Qur'an also tells every sinner not to despair, because Allah (﷾) forgives all sins to those who turn back to Him. Muslims do offer an animal at Eid al-Adha, the festival of the sacrifice. But the Qur'an says it is not their meat or their blood that reaches Allah (﷾), only their piety (22:37).
“Yet now, if thou wilt forgive their sin—; and if not, blot me, I pray thee, out of thy book which thou hast written. And the Lord said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.”
“The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.”
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
“Say: "O my Servants who have transgressed against their souls! Despair not of the Mercy of Allah: for Allah forgives all sins: for He is Oft-Forgiving, Most Merciful.”
“It is not their meat nor their blood, that reaches Allah: it is your piety that reaches Him: He has thus made them subject to you, that ye may glorify Allah for His Guidance to you and proclaim the good news to all who do right.”
Verse by verse
Hebrews 9:22 — Epistle to the Hebrews, chapter 9, verse 22
“And almost all things are by the law purged with blood; and without shedding of blood is no remission.”
Cited for
Christians read this as a universal rule: without shed blood there is no forgiveness of sin at all.
The response
The writer himself says only "almost all things" were cleansed with blood. The Law he describes forgave a poor man's sin with a handful of flour burned on the altar, and says "it shall be forgiven him". So the Law itself did not make blood the only way.
Where it lands — By the Law's own words, blood was the usual way to forgiveness, not the only way.
Leviticus 17:11 — Torah, Vayikra, chapter 17, verse 11
“For the life of the flesh is in the blood: and I have given it to you upon the altar to make an atonement for your souls: for it is the blood that maketh an atonement for the soul.”
Cited for
Christians read this as proof that God accepts no atonement except blood.
The response
The verse explains why Israel must not eat blood: it belongs to the altar. It does not say blood is the only way back to God (﷾). The same book forgives with flour, and the prophets forgive on repentance alone. Jesus himself quoted "I will have mercy, and not sacrifice".
Where it lands — The verse explains the ban on eating blood. The Bible elsewhere shows God (﷾) forgiving with no blood at all.
Romans 3:25 — Epistle to the Romans, chapter 3, verse 25
“Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;”
Cited for
Christians read this as saying that those forgiven before the cross were forgiven in advance, because of the blood Jesus would shed.
The response
Neither the book of Jonah nor Jesus says so. Each gives its own reason. Nineveh was spared because "they turned from their evil way". The tax collector was forgiven because "he that humbleth himself shall be exalted". Neither mentions a future sacrifice.
Where it lands — The texts give repentance and humility as the reason for forgiveness, not a sacrifice still to come.
Matthew 26:28 — Gospel of Matthew, chapter 26, verse 28
“For this is my blood of the new testament, which is shed for many for the remission of sins.”
Cited for
Christians read this as Jesus himself saying that sins are forgiven through his blood.
The response
Jesus' other words tie forgiveness to something else. He told a paralysed man, "Son, thy sins be forgiven thee", and no sacrifice was offered. He taught that God (﷾) forgives those who forgive others. In his parable the tax collector asked only for mercy and went home justified. And the Qur'an says of Jesus, "they killed him not, nor crucified him" (4:157).
Where it lands — In Jesus' own words, forgiveness comes through faith, mercy and forgiving others, and the Qur'an denies the death this reading needs.
Perspectives
Mainstream Christian
Christians say the Law taught that sin is forgiven through blood. They hold that the animal sacrifices pointed forward to the death of Jesus, whose blood takes away sin once for all.
Hebrews 9:22 says that without the shedding of blood there is no remission.
Rebuttal
The verse speaks of the Law. But the Law itself forgave a poor man's sin with a handful of flour burned on the altar: "it shall be forgiven him". The verse itself says only "almost all things", not all, were cleansed with blood.
Leviticus 17:11 says the blood makes atonement for the soul, so forgiveness requires a blood sacrifice.
Rebuttal
The verse explains why Israel must not eat blood: it belongs to the altar. It does not say blood is the only way back to God (﷾). The same book forgives with flour, the prophets forgive on repentance alone, and Jesus quoted "I will have mercy, and not sacrifice".
The Islamic perspective
God (﷾) forgives the one who repents and turns to Him. No blood is needed and no one else is punished. The Hebrew Bible, and Jesus' own teaching on repentance, agree with the Qur'an on this.
The Law of Moses forgave a poor man's sin through an offering of flour, with no blood shed.
Ezekiel promises life and forgiveness to the wicked man who turns from his sins, with no sacrifice named.
God (﷾) spared Nineveh because its people turned from evil, and the text records no sacrifice.
Jesus twice quoted "I will have mercy, and not sacrifice". In his parable the tax collector was justified by asking for mercy alone.
God (﷾) answered Moses that He would blot out only whoever had sinned.
The Qur'an teaches that no soul carries another's sin. Allah (﷾) forgives all sins to those who turn to Him.