The first stanza of "Journey of the Magi" is largely Eliot's creation to tell the story, as he imagines it, of the trip and its conditions. "With the voices singing in our ears" could be interpreted as referring to the choir of angels singing in the night, as described in Luke 2:13.
The second stanza is filled with scriptural references. Jesus described himself as being "living water" (John 7:38, John 4:10-14), paralleling the "running stream" in the poem. The poem's stream beats "the darkness" in the same way that Jesus's life and death defeated the dark of humanity's sin as remembered by the "three trees on the low sky."
"Six hands at an open door dicing" could reference the guards gambling for ownership of Jesus's robe (John 19:24). The "pieces of silver" echoes the payment given to Judas in return for his betrayal of Jesus to the Jewish leaders. The mention of "empty wine-skins" recalls the parable about putting new wine into old wine-skins (Matthew 9:16-17).
The final stanza, the reflections of the narrator years later, contains more general allusions to the scriptural accounts of Jesus's birth and death, including the speaker's puzzlement about the death of Jesus that was "different" from "our death." "No longer at ease here," the narrator now feels disconnected from "an alien people clutching their gods" and looks forward to "another death" that will take him away from "the old dispensation." With his death, he will somehow become connected to the new birth he had witnessed many years before, and to the new life after death that had become possible.