A current prayer list can be found here.

Can we as Orthodox Christians pray for non-Orthodox: Living and Reposed?

The answer is, yes.  We know this to be true from examples in Holy Scripture, as well as those given to us by Saints of our Church.  According to Tradition, however, these prayers must be in the privacy of our own rooms or prayer corners at home.  These prayers are not made during the corporate offering of the Divine Liturgy or Daily Office. There is, of course, the modern alteration of petitions in our services for "Our God-fearing Emperor" to say "The President of our Country," who is usually not an Orthodox Christian, but this alteration is not universal and should not be seen as the criterion for opposing Tradition. There is also the practice of offering a Molieben or Akathist service for particular occasions or needs that may extend beyond the Eucharistic gathering, appealing for God's mercy and help.

Question: "But how is one to pray for such persons?"

Answer: "In the spirit of the virtuous and wise."  Pray simply, without inquiring, entrusting your heart to the right hand of the All-high. 

Along with St. Macarius of Egypt, the Optina Elder Leonid, who was experienced in the spiritual life, can serve Orthodox Christians as an example or paradigm of prayer for some non-Orthodox persons close to us.  We don't know (and to no one of us has it been revealed) what or how much benefit such a prayer can bring to the non-Orthodox.  The more humble and self-abasing the prayer, the more hopeful and beneficial.

Elder Leonid left us the following prayers for the living and reposed non-Orthodox.

Prayer for Praying for Non-Orthodox (Living)

Have mercy, O Lord, if it is possible, on the souls of Thy servants (include the names) who are separated from Thy Holy Orthodox Church!  Unfathomable are Thy judgments. Do not account this prayer of mine as sin. But, may Thy Holy Will be done!

Prayer for Praying for Non-Orthodox (Reposed)

Have mercy, O Lord, if it is possible, on the souls of Thy servants, departed this life in separation from Thy Holy Orthodox Church!  Unfathomable are Thy judgments.  Do not account this prayer of mine as sin.  But, may Thy Holy Will be done!

It has also been taught that during the corporate offering of the Lord's Prayer, when in humble submission we pray, "Thy Kingdom come, Thy will be done on earth, as it is in heaven," we may have in mind those needs that are outside the Eucharistic gathering, consigning them to the will of God and not the demands of our own will.  This could be why the Elder Leonid would have us to recognize the potential for our request to ask God to conform to our will by including in his prayer for the Non-Orthodox, "Do not account this prayer of mine as sin." Whether praying for the Orthodox or Non-Orthodox, we must all come to the point of accepting what happens as being within God's will, even if we must trust God to make the evil that befalls us or others to be good by His goodness (Anaphora in the Divine Liturgy of St. Basil the Great).

While this teaching could seem to be a strict limitation, it may be helpful to consider it as a "middle way."  The extremes would be represented by the teaching not to pray for anyone outside of the Orthodox Church (extremely exclusive), and the teaching that allows any person to be commemorated during the Liturgy of Preparation or Divine Liturgy, which would be alarmingly close to allowing any person to receive Holy Communion regardless of their relation to the Holy Orthodox Church (extremely inclusive).

-Prepared from sources by Fr. Paul and Fr. David