Objection 1. It would seem that one who is in sin can without sin exercise the order he has received. For since, by virtue of his office, he is bound to exercise his order, he sins if he fails to do so. If therefore he sins by exercising it, he cannot avoid sin: which is inadmissible.
Objection 2. Further, a dispensation is a relaxation of the law. Therefore although by rights it would be unlawful for him to exercise the order he has received, it would be lawful for him to do so by dispensation.
Objection 3. Further, whoever co-operates with another in a mortal sin, sins mortally. If therefore a sinner sins mortally by exercising his order, he who receives or demands any Divine thing from him also sins mortally: and this seems absurd.
Objection 4. Further, if he sins by exercising his order, it follows that every act of his order that he performs is a mortal sin; and consequently since many acts concur in the one exercise of his order, it would seem that he commits many mortal sins: which seems very hard.
On the contrary, Dionysius says (Ep. ad Demophil.): "It seems presumptuous for such a man, one to wit who is not enlightened, to lay hands on priestly things; he is not afraid nor ashamed, all unworthy that he is to take part in Divine things, with the thought that God does not see what he sees in himself; he thinks, by false pretense, to cheat Him Whom he falsely calls his Father; he dares to utter in the person of Christ, words polluted by his infamy, I will not call them prayers, over the Divine symbols." Therefore a priest is a blasphemer and a cheat if he exercises his order unworthily, and thus he sins mortally: and in like manner any other person in orders.
Further, holiness of life is required in one who receives an order, that he may be qualified to exercise it. Now a man sins mortally if he present himself for orders in mortal sin. Much more therefore does he sin mortally whenever he exercises his order.
I answer that, The law prescribes (Deuteronomy 16:20) that "man should follow justly after that which is just." Wherefore whoever fulfills unworthily the duties of his order follows unjustly after that which is just, and acts contrary to a precept of the law, and thereby sins mortally. Now anyone who exercises a sacred office in mortal sin, without doubt does so unworthily. Hence it is clear that he sins mortally.
Reply to Objection 1. He is not perplexed as though he were in the necessity of sinning; for he can renounce his sin, or resign his office whereby he was bound to the exercise of his order.
Reply to Objection 2. The natural law allows of no dispensation; and it is of natural law that man handle holy things holily. Therefore no one can dispense from this.
Reply to Objection 3. So long as a minister of the Church who is in mortal sin is recognized by the Church, his subject must receive the sacraments from him, since this is the purpose for which he is bound to him. Nevertheless, outside the case of necessity, it would not be safe to induce him to an execution of his Order, as long as he is conscious of being in mortal sin, which conscience, however, he can lay aside since a man is repaired in an instant by Divine grace.
Reply to Objection 4. When any man performs an action as a minister of the Church while in a state of mortal sin, he sins mortally, and as often as he performs that action, since, as Dionysius says (Eccl. Hier. i), "it is wrong for the unclean even to touch the symbols," i.e. the sacramental signs. Hence when they touch sacred things in the exercise of their office they sin mortally. It would be otherwise if they were to touch some sacred thing or perform some sacred duty in a case of necessity, when it would be allowable even to a layman, for instance if they were to baptize in a case of urgency, or gather up the Lord's body should it be cast to the ground.
Source: Summa Theologica (New Advent)