Code of Canon Law, canon 938 §5 “The person in charge of a church or oratory is to see to it that the key of the tabernacle in which the blessed Eucharist is reserved, is in maximum safe keeping.” [The Code of Canon Law: New revised English Translation, HarperCollins Liturgical, 1997, ISBN 000599375X.]
From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“317. In no way should all the other things prescribed by law concerning the reservation of the Most Holy Eucharist be forgotten.[footnote 131. Cf. particularly in Sacred Congregation for the Sacraments, Instruction Nullo umquam tempore, 28 May 1938: AAS 30 (1938), pp. 198-207; Codex Iuris Canonici, cann. 934-944.”
The footnoted 1938 Instruction is at
RomanRite - Reservation of the Eucharist . It includes:
"6. (c)
The key of the tabernacle must be most diligently kept by a priest. All the cautions mentioned up to the present will be in vain, if the chief caution, namely, the safe-keeping of the key of the tabernacle be neglected. The fourth paragraph of the above canon expressly mentions in respect to this point that a grave burden rests on the conscience of the priest to whom the key of the tabernacle is entrusted. In order to satisfy this obligation of most diligent custody in regard to the key, the Rector is solemnly warned that the key of the tabernacle must never be left on the table of the altar, nor in the door of the tabernacle, not even at the time when the divine offices are carried out in the morning at the altar of the Blessed Sacrament, or communion is distributed, especially if this altar is not in open view. When these offices are over, the key must be kept by the Rector at home, or always carried about by him, care being taken against losing it; or let it be kept in the sacristy in a safe and secret place, under lock and key, the second key being kept by the Rector as above. [Footnote 6: Cfr. Encycl. litt. iussu Benedicti XIV edit. a S.C.EE. et RR, die 9 Feb. 1751.]
Let priests who are guardians of the B. Sacrament seriously consider that the obligation of keeping most diligently the key of the Sacred Ciborium is a grave obligation, as its scope and the very words of the law clearly show. The priest on whom the right and duty of keeping the key ordinarily and naturally rests is the Rector of the church or oratory; should he go away, he can, and ought, during his absence, entrust the keeping of the Blessed Sacrament to another priest. If he leaves the key in the sacristy under another key; he can give this latter to the sacristan during such time as he is absent, and the key of the tabernacle may be needed. Universal practice is manifestly in favour of this. …"
But it is difficult to reconcile this with the lay people being able to do exposition of the Blessed Sacrament. From Holy Communion and the Worship of the Eucharist Outside of Mass, n. 91, about extraordinary ministers: “Such ministers may open the tabernacle and also, as required, place the ciborium on the altar or place the host in the monstrance. At the end of the period of adoration, they replace the blessed sacrament in the tabernacle.” [The Rites Volume One, Liturgical Press, 1990, ISBN: 0-8146-6015-0, pages 673-674].