Answer
Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Answering your question, Sheikh Ahmad Kutty, a Senior Lecturer and an Islamic Scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
There is no basis for such an irrational belief in Islam. It is at once rejected by the fundamental teachings of Islam even as it amounts to attributing injustice to Allah who introduces Himself to us above all as being Just, Wise and Compassionate.
The Qur’an makes it abundantly clear that each person is responsible for his or her actions and that no soul bears the burden of another. “No soul shall be called to bear the burden of another.” So, how can a child born of Zina be accountable for the sin of his or her father?
The adulterer is responsible for his sin; the child born of this heinous act has no share in it; hence it is born into a state of Fitrah or natural innocence like every other child.
The Prophet (peace be upon him) said, “Every child is born into a state of Fitrah or natural state of innocence, and it is his or her parents that make him a Jew or Christian.”
The Prophet (peace be upon him) used the word ‘every child, ’ and he did not make any exception; so it applies to all children whether they are born in wedlock or outside of it.
Almighty Allah knows best.