The doctrine of Original Sin being addressed in this article is not the Biblical teaching that all human beings have, as a result of their flesh born after Adam, a proclivity to sin - sometimes called a "sin nature", and are therefore unable to justify themselves on the basis of works. This doctrine, covered in this article, is taught throughout the New Testament, and is not really controversial.
The doctrine that this article is addressing is the dogma that all human beings personally bear the original sin that Adam committed in the Garden of Eden, and are condemned before God on that basis. Therefore, everyone - including infants, and the mentally ill - are by default condemned by God, even if they have not committed any actual sin.
There are many verses in the Bible which teach that all have sinned (Romans 3:10-12, 3:23, 5:12, Psalm 14:2-3, Galatians 3:22, Ecclesiastes 7:20: 1 Kings 8:46).
One of these passages in particular is seen as the bedrock of the doctrine of Original Sin by its proponents, that being the one from Romans chapter five:
Romans 5:12
12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
The above passage can be used as a proxy to discuss all of the other verses that speak about "all" sinning, since it ultimately teaches the same thing that they all teach.
Firstly, what is a "sin"? This is clearly defined in Scripture as "the transgression of the law":
1 John 3:4
4 Whosoever committeth sin transgresseth also the law: for sin is the transgression of the law.
Which law? Both natural law, given to all of mankind in the form of a conscience (Romans 2:14-15), along with the Mosaic Law, which was added to increase transgressions (Romans 5:20), thereby driving us faster toward the Savior Jesus (Galatians 3:24).
So, does an infant sin? Does a mentally ill person sin? The answer is, they may commit actions that are against the Law of God, but that sin is not imputed to them, as they are innocent in the same manner that Adam and Eve were before the fall:
Deuteronomy 1:39
39 Moreover your little ones, which ye said should be a prey, and your children, which in that day had no knowledge between good and evil, they shall go in thither, and unto them will I give it, and they shall possess it.
Isaiah 7:16
16 For before the child shall know to refuse the evil, and choose the good, the land that thou abhorrest shall be forsaken of both her kings.
When a child does not know right from wrong, they do not have a moral conscience. This means they are not condemned by the natural law given to all mankind (Romans 2:14-15), as "sin is not imputed where there is no law":
Romans 5:12-13
12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
13 For until the law sin was in the world: but sin is not imputed when there is no law.
Above is in context to the Mosaic Law, and essentially teaches that mankind sinned by that Law before it was given, yet because it had not yet been given, it was not imputed to them as sin. I am extending this to situation of infants in regard to the natural law given to all men, which I think is reasonable, because in both cases - general mankind prior to the Mosaic Law, and infants who do not yet have a conscience - there is no way for them to know what that Law demanded.
This explains how the Apostle Paul was "alive without the Law once":
Romans 7:9
9 For I was alive without the law once: but when the commandment came, sin revived, and I died.
The Apostle Paul, as a child in innocence, was "alive". Then, when the Law became known to him, and he sinned anyway, his spirit "died" (Ephesians 2:1, 2:5, Colossians 2:13). The first sin that you commit against your conscience, or the revealed Law of God, with understanding, is the sin that deadens your spirit. Prior to that, all children are spiritually alive.
This is why when David's infant child dies, he states "I shall go to him" (2 Samuel 12:23). He said so, because the baby was in Heaven with the Lord. This also explains why Jesus Christ repeatedly pointed to children as examples of those in the Kingdom of Heaven (Matthew 11:25, 18:3, 19:13-14, Mark 9:35-37, 10:13-15, Luke 9:46-48, 10:21, 18:15-17).
So, returning to Romans 5:12:
Romans 5:12
12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
The "death" mentioned here is physical, and spiritual.
The physical death is imputed to all of mankind (1 Corinthians 15:21-22, Romans 5:14-15) by virtue of being born after Adam, regardless of whether they have had any sin actually imputed to them yet, which is why infants, and even babies in the womb, die. Being born in the lineage of Adam also gives everyone a sin nature, whereby they tend toward sin, which is what verses like Psalm 51:5, Ephesians 2:3, 4:22, Romans 5:19, 6:6, and so on, refer to.
The spiritual death is imputed upon the first imputed sin, which is the first sin that someone does outside of an innocent state.
So, when the Bible says "all have sinned", it is a general statement that this true for virtually all of mankind, excepting those who die before breaking the Law of God, which is the definition of sin. Yet it does not mean that "all are damned", because sin is not imputed to innocents.
In light of the above, the Bible absolutely does not teach that infants, and the mentally ill, will be condemned on the basis of a sin that Adam committed. In fact, it actively argues against such a ridiculous, unjust dogma:
Ezekiel 18:20
20 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.
The above states that a child does not bear their parent's sins, directly countering the false doctrine of Original Sin, which argues that God has imposed this injustice on every person. This point is reiterated in Jeremiah 31:29-30, Deuteronomy 24:16, and Ezekiel 18:2-4 - four times we are admonished of this principle of justice, which is supremely just, and logical.
And that really gets to the heart of this false doctrine, and why it needs to be refuted. Intuitively, it is obviously false. There is not a single human being on earth that doesn't instinctively know that infants and the mentally ill being sent to Hell - for something that they did not even do, no less - would be a horrific, nonsensical abomination of justice.
"The wages of sin is death" (Romans 6:23). The wages of birth is not death. Prior to committing a sin with understanding, every child is spiritually alive, though they have a fleshly body, born after Adam. This body ensures their physical death, and should they live to maturity, also ensures that they will eventually break the Law of God with understanding, which will result in imputed sin, and consequently, their spiritual death.
However, it's an error to teach that all of mankind, including infants and the mentally ill, bear personal guilt before God, and therefore deserve condemnation, for the original sin of Adam.