Imam Sadiq (as) has been reported to have said,
An illegitimate child is just like any other person. He will be rewarded for a good deed and will be punished for a bad deed.
There is no doubt that an illegitimate child bears no guilt concerning the sin his parents committed. God says,
Nor can a bearer of burdens bear another's burdens.1
Imam Sadiq (as) has been reported to have said,
An illegitimate child is just like any other person. He will be rewarded for a good deed and will be punished for a bad deed.2
In answer to why such children do not inherit from their parents, it should be said that Islam has established the law of inheritance for people who have certain legal relationships with one other. An illegitimate child does not have any legal relationship with his adultererous parents since their relationship had not been legal. Therefore, according to the narrations, neither the illegitimate child inherits from his parents nor the parents inherit from him.3
In addition, this rule is a mechanism to further help curb the heinous act of prostitution because anyone who might be tempted to commit this grave sin, when considers the negative consequences of such relationship, would be discouraged to commit it.
Of course, the adulterer (father) is obliged to pay child support and meet the needs of his illegitimate child. If this is not fulfilled, the Islamic government must take care of the child so that there is no pressure on the illegitimate child.
References:
1. Quran, 35:18.
2. Al-Kafi, Vol. 8, p. 238.
3. Wasa’il Al-Shi’ah, vol. 26, p. 274.