Glory to Jesus Christ!
Deacon Paul is right. Traditionally, the liturgical funerary acclamation "Eternal memory!" is used only for Orthodox Christians who have died.
Many people misunderstand this phrase. It is not a call for the dead to remain alive forever in *our* memory, but rather a prayer that he, she or they will remain alive forever in *God's* memory, as Protopresbyter Thomas Hopko explains in his four-volume series of books, "The Orthodox Faith" (Volume 2: Worship, "The Sacraments," "Funeral"):
"It has to be noted here that this song, contrary to the common understanding of it, is the supplication that God would remember the dead, for in the Bible it is God's 'eternal memory' which keeps man alive. *Sheol* or *Hades* or the *Pit,* the biblical realm of the dead also called *Abaddon,* is the condition of forsakenness and forgottenness by God. It is the situation of non-life since in such a condition no one can praise the Lord; and the praise of the Lord is the only content and purpose of man's life; it is the very reason for his existence. Thus, this most famous and final of the Orthodox funeral hymns is the prayer that the departed be eternally alive in the 'eternal rest' of the 'eternal memory' of God -- all of which is made possible and actual by the resurrection of Jesus Christ which is the destruction of the Pit of Death by the splendor of Divine Righteousness and Life (see Psalm 88, Hosea 13:14, 1 Corinthians 15; Ephesians 4:9; Philippians 2:5-11; 1 Peter 3)."
As Saint Theophanes the Recluse noted, it is not possible for us to have communion in death with those with whom we did not have it in life. For this reason, we do not commemorate by name those who are not or were not Orthodox Christians in the public prayers of the Church, or in the proskomide of the Divine Liturgy, though we are certainly encouraged to remember them in our personal prayers, offerings of candles and vigil lamp oil, and alms, asking God to grant them a just reward on Judgment Day, since Saint Paul the Apostle tells us that he will judge those outside the Church on the basis of the "law written on their hearts," even though they did not have "the law" of the Church's Orthodox Christianity (Romans 2:14-16).
"Lord, have mercy!" is always appropriate in any situation, heterodox or Orthodox. But traditionally, "Eternal memory!" is reserved for Orthodox Christians, as its use in the Sunday of Orthodoxy service (as contrasted with the mention of the heterodox heresiarchs) demonstrates.
With prayers and good will,
Gregory Orloff