Objection 1. It would seem that a prince does not so forfeit his dominion over his subjects, on account of apostasy from the faith, that they no longer owe him allegiance. For Ambrose [St. Augustine, Super Psalm 124:3 says that the Emperor Julian, though an apostate, nevertheless had under him Christian soldiers, who when he said to them, "Fall into line for the defense of the republic," were bound to obey. Therefore subjects are not absolved from their allegiance to their prince on account of his apostasy.
Objection 2. Further, an apostate from the faith is an unbeliever. Now we find that certain holy men served unbelieving masters; thus Joseph served Pharaoh, Daniel served Nabuchodonosor, and Mardochai served Assuerus. Therefore apostasy from the faith does not release subjects from allegiance to their sovereign.
Objection 3. Further, just as by apostasy from the faith, a man turns away from God, so does every sin. Consequently if, on account of apostasy from the faith, princes were to lose their right to command those of their subjects who are believers, they would equally lose it on account of other sins: which is evidently not the case. Therefore we ought not to refuse allegiance to a sovereign on account of his apostatizing from the faith.
On the contrary, Gregory VII says (Council, Roman V): "Holding to the institutions of our holy predecessors, we, by our apostolic authority, absolve from their oath those who through loyalty or through the sacred bond of an oath owe allegiance to excommunicated persons: and we absolutely forbid them to continue their allegiance to such persons, until these shall have made amends." Now apostates from the faith, like heretics, are excommunicated, according to the Decretal [Extra, De Haereticis, cap. Ad abolendam]. Therefore princes should not be obeyed when they have apostatized from the faith.
I answer that, As stated above (Question 10, Article 10), unbelief, in itself, is not inconsistent with dominion, since dominion is a device of the law of nations which is a human law: whereas the distinction between believers and unbelievers is of Divine right, which does not annul human right. Nevertheless a man who sins by unbelief may be sentenced to the loss of his right of dominion, as also, sometimes, on account of other sins.
Now it is not within the competency of the Church to punish unbelief in those who have never received the faith, according to the saying of the Apostle (1 Corinthians 5:12): "What have I to do to judge them that are without?" She can, however, pass sentence of punishment on the unbelief of those who have received the faith: and it is fitting that they should be punished by being deprived of the allegiance of their subjects: for this same allegiance might conduce to great corruption of the faith, since, as was stated above (1, Objection 2), "a man that is an apostate... with a wicked heart deviseth evil, and... soweth discord," in order to sever others from the faith. Consequently, as soon as sentence of excommunication is passed on a man on account of apostasy from the faith, his subjects are "ipso facto" absolved from his authority and from the oath of allegiance whereby they were bound to him.
Reply to Objection 1. At that time the Church was but recently instituted, and had not, as yet, the power of curbing earthly princes; and so she allowed the faithful to obey Julian the Apostate, in matters that were not contrary to the faith, in order to avoid incurring a yet greater danger.
Reply to Objection 2. As stated in the article, it is not a question of those unbelievers who have never received the faith.
Reply to Objection 3. Apostasy from the faith severs man from God altogether, as stated above (Article 1), which is not the case in any other sin.
Source: Summa Theologica (New Advent)