Question: How do you pray for the Non-Orthodox?
We do not pray for the Non-Orthodox by name, in the services of the Church, but we can and should pray for them.
Prayers for the Living
When praying for the Non-Orthodox who are living, you can use this prayer:
"Grant, O Lord, that thy servants will be illumined with the light of the Orthodox Faith, and numbered with Thy one, Holy, Catholic, and Apostolic Church..." and then commemorate their names.
Some use a form of the Jesus Prayer to pray for others, which could include the Non-Orthodox: "O Lord, Jesus Christ, Son of God, have mercy on N."
You can also pray in your own words for the Non-Orthodox. Some think that it is wrong for Orthodox Christians to pray extemporaneously, but this is not true. The Jordanville Prayerbook specifically says, towards the end of the morning prayers:
"Then offer a brief prayer for the health and salvation of thy spiritual father, thy parents, relatives, those in authority, benefactors, and others known to thee, the ailing, or those passing through sorrows."
When we do pray extemporaneously, however, we should pray in an Orthodox manner... which we learn how to do by praying the prayers of the Church.
Prayers for the Departed
You can use this prayer, "Have mercy, O Lord, if it be possible on the souls of Thy departed servants (names), who have departed into eternal life in separation from Thy Holy Orthodox Church: unsearchable are Thy decrees. Do not account this, my prayer as a sin, but may Thy holy will be done."
This prayer is from St. Leonid of Optina, and is based on a prayer he recommended to a disciple who was in despair over the suicide of his father: "Seek out, O Lord, the lost soul of my father, if possible, have mercy! Thy judgements are unfathomable. Do not account this prayer of mine as a sin. May Thy holy will be done."
Another, more elaborate way to pray for the Non-Orthodox departed (which can be used, privately, for anyone who has departed) is The Akathist for the Repose of the Departed, which is found in The Book of Akathists," published by Holy Trinity Monastery. This is a beautiful akathists, which not only provides a means to pray for those for whom a Pannikhida cannot be done, but also brings great consolation to the one who prayers it.
For more information see:
Praying for the Non-Orthodox
How to Set Up a Personal Commemoration Book
On the Burial of the Heterodox
Will the Heterodox Be Saved?