“Lift up your heart” in the Mass comes from the Book of Revelation. John probably was referencing Lamentation 3:41. There are hundreds of references to the Old Testament in Revelation.
The Book of Revelation roughly follows the liturgy of the Mass, and the part in the Mass where the priest says “Lift up your hearts” corresponds to Revelation 4:1.
After this I looked, and lo, in heaven an open door! And the first voice, which I had heard speaking to me like a trumpet, said, "Come up hither, and I will show you what must take place after this." (Revelation 4:1)
When the priest says, “Lift up your hearts” and we respond “We lift them up to the Lord” - this is where the liturgy taking place 24x7 in heaven is joined with our Earthly liturgy. We “come up hither” as we enter through the same open door which John saw in his vision. We worship together with all the saints and the angels as we say the thrice holy…
And the four living creatures, each of them with six wings, are full of eyes all round and within, and day and night they never cease to sing, "Holy, holy, holy, is the Lord God Almighty, (Revelation 4:8)
The Sanctus (thrice holy) in the Mass corresponds to Revelation 4:8. Our physical and spiritual posture reflects the posture of the 24 elders who worship God.
the twenty-four elders fall down before him who is seated on the throne and worship him who lives for ever and ever; they cast their crowns before the throne, singing, (Revelation 4:10)
Our physical posture - kneeling - reflects (or should reflect) how the 24 elders “fall down before him who is seated on the throne.” Our spiritual posture reflects how the 24 elders “cast their crowns before the throne” as we place all our needs, troubles, gifts and blessings before the Lord who is about the be enthroned on the altar. The Memorial Acclamation mirrors Revelation 4:11.
The whole liturgy and especially those parts around the preface prayer (preface dialogue, memorial acclamation, sanctus, etc) mirrors the Book of Revelation. I imagine myself being lifted up into heaven to join the heavenly liturgy with all the angels and saints when the priest says, “Lift up your hearts.”
-Tim-