Islam does not accept the belief of 'original sin'. Its main concern is the 'acquired sin' and not the 'original sin'. According to Islam 'sin' is neither a hereditary disease that is transferred from a father to his son through the reproductive system nor is sin like a rank or a title that can be passed from an older to a younger person of the family. We can be said to have committed a 'sin' only when we ourselves perpetrate some unlawful, unclean, or prohibited act actually and physically; being in full control of our mental capacity; without any external compulsion or pressure; with our free will; and with our own independent guilty intention: or when we refrain from or escape some lawful duty or obligation. It can also be interpreted as 'disobedience to God's Commandments'. There is a maxim of the legal parlance: 'An act does not make a man guilty unless there be guilty intention'. To some extent it covers the theme.
Guiltless Birth of a Child according to Islam
A human being is born without any sin (guiltless or innocent) and he remains such unless he intentionally commits a sin (i.e. disobeys God's commandments). The Prophet of Islam (sws) is reported to have said, 'Every child is born on fitrah ie. 'nature'or 'norm'. It is his parents that corrupt him'. There is another tradition of the Prophet of Islam (sws) in Musnad Ah~mad:
Of course, my Sustainer says: 'No doubt I created My servants 'H~unafa'a' (ie. men of pure faith), all of them, and then came to them the devils; so they led them astray from their pure faith, and they made upon them unlawful which I had made lawful for them, and they ordered them to make companions with Me for which there exists no logic or evidence'.
The Qur'an says:
So set your face to the religion, a man of pure faith—God's original upon which He originated mankind. There is no changing God's creation. That is the right religion; but most men know it not'.
A renowned commentator of the Qur'an of the twentieth century, Imam Amin Ah~san Is~lah~i, has well explained these verses:
The Almighty Allah has created humankind in the best design and best nature. He bestowed upon him the capacity of discriminating between vice and virtue, right and wrong, and endowed him with the urge for adopting virtue and eschewing vice. But this human nature is not like the instinct of the animals that a human being may not turn away from it. He rather has a choice of his own. That's why he sometimes becomes so blind in pursuit of his materialistic ambitions and worldly attractions that in spite of having complete sense of right and wrong, he not only follows the wrong but also formulates theories in its favour.
What is then the rationale of the revealed guidance? It is not because he had no sense of distinguishing between right and wrong. It is rather because he was prone to some misconceptions due to his shortcomings. Moreover, he did not have the capacity of understanding all the accessories and repercussions of the basic principles of nature. So the Almighty Allah had to raise up messengers among them for their proper guidance. The teachings of these messengers are based on the fundamentals that have been entrusted to human beings. So those who were of pure and undefiled nature, took the teachings of the messengers as the voice of their own conscience [stress added]. But those who had defiled their nature, stood against it. Of course, in the heart of their hearts, they also conceded that the messengers were true. It is due to this that the Qur'an has been called 'Reminder', because it reminds us of our latent, innate and inherent data to which we willingly have turned a deaf ear.
There is an in-built mechanism in human beings to discriminate and differentiate between right and wrong and to choose the right or wrong way of life for themselves. The Qur'anasserts:
Verily we have created man of the mingled seed of both sexes that we might test him: and we have made him to hear and to see. We have surely directed him in the way; whether he be grateful or ungrateful.
In its 91st chapter named 'The Sun', the Qur'an says that Allah has inspired the human soul [the significance of] sin and godliness:
[I swear] by the soul and Him who balanced it, And breathed into it its wickedness and piety, Blessed now is he who hath kept it pure, And undone is he who corrupted it!
A similar theme has been asserted in chapter 90th of the Qur'an which is as follows:
And We have guided him on the two paths? But he has not attempted the steep! What has let you know what is the steep? Setting free of bondmen, or feeding on a day of famine An orphan of kin, Or a destitute [person] downtrodden. And then has become one of those who have believed and who counsel each other to endurance and to compassion.
Under the present heading we have so far seen that:
1. According to Islam every child is born free of sin.
2. It has been endowed with the capacity to distinguish between right and wrong.
3. Allah has managed to strengthen this capacity (conscience) externally through His messengers and the scriptures. It is to afford him exact, unambiguous, and indubious guidance.
4. Even then if someone chooses to commit a sin of his own free will, he himself is answerable for that. (As the Bible says, 'God will require it of him'.)
Natural Justice Demands that Everyone be Answerable for his Own Wrongs and not for Others' Misdeeds
It is recorded in Jeremiah that everyone will be meted out retribution for his own sins. It simply means that nobody can bear the burden of others' crimes.
When that time comes, people will no longer say, 'The parents ate the sour grapes, But the children got the sour taste.' Instead whoever eats sour grapes will have his own teeth set on edge; and everyone will die because of his own sin. (…) I will forgive their sins and I will no longer remember their wrongs. I, the Lord, have spoken.
The book of Deuteronomy asserts that father shall not be held responsible for the son's sins and vice versa:
Fathers shall not be put to death for their children, nor children for their fathers; only for his own guilt shall a man be put to death.
Isaiah also reiterates the theme in the following words,
Woe to them! They have brought disaster upon themselves. Tell the righteous it will be well with them, for they will enjoy the fruit of their deeds. Woe to the wicked! Disaster is upon them! They will be paid back for what their hands have done.
2Kings, XIV: 6, while reproducing this verse, changes the last clause as: 'each one shall die of his own sin.'
The same theme has been asserted by Ezekiel in fair detail. I have inserted some brief footnotes to it on the spot. It would be advisable that they be studied carefully side by side with the text to understand the theme properly:
The Lord spoke to me and said: 'What is this proverb people keep repeating in the land of Israel? 'The parents ate the sour grapes, But the children got the sour taste [stress added].' "As surely as I am the living God," says the Sovereign Lord, "You will not repeat the proverb in Israel any more. The life of every person belongs to me, the life of the parent as well as that of the child. The person who sins is the one who will die [stress added].
"Suppose there is a truly good man, righteous and honest. He doesn't worship the idols of the Israelites or eat the sacrifices offered at forbidden shrines. He doesn't seduce another man's wife or have intercourse with a woman during her period. He doesn't cheat or rob anyone. He returns what a borrower gives him as security; he feeds the hungry and gives clothing to the naked. He doesn't lend money for profit. He refuses to do evil and gives an honest decision in any dispute. Such a man obeys my commands and carefully keeps my laws. He is righteous, and he will live," says the Sovereign Lord.
"Then suppose this man has a son who robs and kills, who does any of these things that the father never did. He eats sacrifices offered at forbidden shrines and seduces other men's wives. He cheats the poor, he robs, he keeps what a borrower gives him as security. He goes to pagan shrines, worships disgusting idols, and lends money for profit. Will he live? No, he will not. He has done all these disgusting things, and so he will die. He will be to blame for his own death.
"Now suppose this second man has a son. He sees all the things his father practised, but does not follow his example. He doesn't worship the idols of the Israelites or eat the sacrifices offered at forbidden shrines. He doesn't seduce another man's wife or oppress any one or rob any one. He returns what a borrower gives him as security. He feeds the hungry and gives clothing to the naked. He refuses to do evil and doesn't lend money for profit. He keeps my laws and obeys my commands. He will not die because of his father's sins [stress added], but he will certainly live. His father, on the other hand, cheated and robbed, and always did evil to every one. And so he died because of the sins he himself had committed [stress added].
"But you ask: Why shouldn't the son suffer because of his father's sins? The answer is that the son did what was right and good. He kept my laws and followed them carefully, and so he will certainly live. It is the one who sins who will die. A son is not to suffer because of his father's sins, nor a father because of the sins of his son [stress added]. A good man will be rewarded for doing good, and an evil man will suffer for the evil he does. 
"If an evil man stops sinning and keeps my laws, if he does what is right and good, he will not die; he will certainly live. All his sins will be forgiven, and he will live, because he did what is right. Do you think I enjoy seeing an evil man die?'' asks the Sovereign Lord. "No, I would rather see him repent and live. 
"But if a righteous man stops doing good and starts doing all the evil, disgusting things that evil men do, will he go on living? No! None of the good he did will be remembered. He will die because of his unfaithfulness and his sins.
"But you say, 'What the lord does isn't right.' Listen to me, you Israelites. You think my way of doing things isn't right? It is your way that isn't right. When a righteous man stops doing good and stops doing evil and then dies, he dies because of the evil he has done. When an evil man stops sinning and does what is right and good, he saves his life. He realizes what he is doing and stops sinning, so he will certainly not die, but go on living. (…).
"Now I, the Sovereign Lord, am telling you Israelites that I will judge each of you by what he has done. Turn away from all the evil you are doing, and don't let your sin destroy you. Give up all the evil you have been doing, and get yourselves new minds and hearts. Why do you Israelites want to die? I do not want anyone to die [stress added], " says the Sovereign Lord. "Turn away from your sins and live." 
Jesus did not come to change the 'Law'
As regards the NT, first of all it is to be noticed that Jesus never and nowhere claimed to change, cancel, or abrogate the OT. He rather exclusively and unequivocally asserted, quite contrary to it as recorded in the Gospel according to Matthew:
"Do not think that I came to abolish the Law or the Prophets; I did not come to abolish, but to fulfil. "For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass away from the Law; until all is accomplished. Whoever then annuls one of the least of these commandments, and so teaches others [stress added], shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven.
Similar words have been repeated in the Gospel According to Luke as well:
"The Law and the Prophets were proclaimed until John; since then the gospel of the kingdom of God is preached, and everyone is forcing his way into it. "But it is easier for heaven and earth to pass away than for one stroke of a letter of the Law to fail.
It is Incumbent on Everyone to Follow the 'Law'
It is recorded in the Gospel according to Mark that people should follow the Law of God and not the man-made rules:
'(…). Vainly they worship me, when teaching human regulations as doctrines.' You let go of God's commandments to cling to human tradition." He added, "How well you frustrate the Law of God to observe your own tradition'. 
Matthew has noted the similar theme in his gospel in these words:
Why do ye also transgress the commandment of God by your tradition? (…). Thus have ye made the commandment of God of none effect by your tradition. Ye hypocrites, Well did Esaias prophesy of you, saying, This people draweth nigh unto me with their mouth, and honoureth me with their lips; but their heart is far from me. But in vain they do worship me, teaching for doctrines the commandments of men.
Moses has cursed those who do not act according to the 'Law', as is reproduced below:
'Cursed is he who does not confirm the words of this law by doing them.' And all the people shall say, 'Amen.'
Man is Born Free of Sin (and not with some Original Sin)
It is recorded in the Gospel of Mark that Jesus believed the children to be born free of sin and it is they to whom belongs the Kingdom of God:
Some people brought children to Jesus for him to place his hands on them, but the disciples scolded the people. When Jesus noticed this, he was angry and said to his disciples, "Let the children come to me, and do not stop them, because the kingdom of God belongs to such as these. I assure you that whoever does not receive the Kingdom of God like a child will never enter it [stress added]." Then he took the children in his arms, placed his hands on each of them, and blessed them.
Matthew has also recorded this theme in his gospel at two different places. One of them is:
At that time the disciples came to Jesus, asking, "Who is the greatest in the Kingdom of heaven?" So Jesus called a child, made him stand in front of them, and said, I assure you that unless you change and become like children, you will never enter the Kingdom of heaven. The greatest in the Kingdom of heaven is the one who humbles himself and becomes like this child [stress added]. And whoever welcomes in my name one such child as this, welcomes me.
Luke has recorded a similar event in his gospel:
People even brought babies to him, for him to touch them; but when the disciples saw this they scolded them. But Jesus called the children to him and said, 'Let the little children come to me, and do not stop them; for it is to such as these that the Kingdom of God belongs. In truth I tell you, anyone who does not welcome the Kingdom of God like a little child will never enter it.
The above evidence makes it abundantly clear that according to Jesus children are innocent and free from sin.