Objection 1. It would seem that those who are of illegitimate birth should not be debarred from receiving Orders. For the son should not bear the iniquity of the father (Ezekiel 18:20); and yet he would if this were an impediment to his receiving Orders. Therefore, etc.
Objection 2. Further, one's own fault is a greater impediment than the fault of another. Now unlawful intercourse does not always debar a man from receiving Orders. Therefore neither should he be debarred by the unlawful intercourse of his father.
On the contrary, It is written (Deuteronomy 23:2): "A mamzer, that is to say, one born of a prostitute, shall not enter into the Church of the Lord until the tenth generation." Much less therefore should he be ordained.
I answer that, Those who are ordained are placed in a position of dignity over others. Hence by a kind of propriety it is requisite that they should be without reproach, not for the validity but for the lawfulness of the sacrament, namely that they should be of good repute, bedecked with a virtuous life, and not publicly penitent. And since a man's good name is bedimmed by a sinful origin, therefore those also who are born of an unlawful union are debarred from receiving orders, unless they receive a dispensation; and this is the more difficult to obtain, according as their origin is more discreditable.
Reply to Objection 1. Irregularity is not a punishment due for sin. Hence it is clear that those who are of illegitimate birth do not bear the iniquity of their father through being irregular.
Reply to Objection 2. What a man does by his own act can be removed by repentance and by a contrary act; not so the things which are from nature. Hence the comparison fails between sinful act and sinful origin.
Source: Summa Theologica (New Advent)