Objection 1. It would seem that the priests of the Law had the keys. For the possession of the keys results from having orders. But they had orders since they were called priests. Therefore the priests of the Law had the keys.
Objection 2. Further, as the Master states (Sent. iv, D, 18), there are two keys, knowledge of discretion, and power of judgment. But the priests of the Law had authority for both of these: therefore they had the keys.
Objection 3. Further, the priests of the Law had some power over the rest of the people, which power was not temporal, else the kingly power would not have differed from the priestly power. Therefore it was a spiritual power; and this is the key. Therefore they had the key.
On the contrary, The keys are ordained to the opening of the heavenly kingdom, which could not be opened before Christ's Passion. Therefore the priest of the Law had not the keys. Further, the sacraments of the old Law did not confer grace. Now the gate of the heavenly kingdom could not be opened except by means of grace. Therefore it could not be opened by means of those sacraments, so that the priests who administered them, had not the keys of the heavenly kingdom.
I answer that, Some have held that, under the Old Law, the keys of the kingdom were in the hands of the priests, because the right of imposing punishment for sin was conferred on them, as related in Leviticus 5, which right seems to belong to the keys; but that these keys were incomplete then, whereas now they are complete as bestowed by Christ on the priests of the New Law.
But this seems to be contrary to the intent of the Apostle in the Epistle to the Hebrews (Hebrews 9:11-12). For there the priesthood of Christ is given the preference over the priesthood of the Law, inasmuch as Christ came, "a high priest of the good things to come," and brought us "by His own blood" into a tabernacle not made with hand, whither the priesthood of the Old Law brought men "by the blood of goats and of oxen." Hence it is clear that the power of that priesthood did not reach to heavenly things but to the shadow of heavenly things: and so, we must say with others that they had not the keys, but that the keys were foreshadowed in them.
Reply to Objection 1. The keys of the kingdom go with the priesthood whereby man is brought into the heavenly kingdom, but such was not the priesthood of Levi; hence it had the keys, not of heaven, but of an earthly tabernacle.
Reply to Objection 2. The priests of the Old Law had authority to discern and judge, but not to admit those they judged into heaven, but only into the shadow of heavenly things.
Reply to Objection 3. They had no spiritual power, since, by the sacraments of the Law, they cleansed men not from their sins but from irregularities, so that those who were cleansed by them could enter into a tabernacle which was "made with hand."
Source: Summa Theologica (New Advent)