The Divine Liturgy is a Sacrament and is for the purpose of allowing the People of the Orthodox Church to offer praise and worship of the Most Holy Trinity. It is a time to commune with the Most Holy Trinity. As it is a Sacarment reserved for the Orthodox Christian, we do not pray by name for the non-orthodox. Catechumen are part of the Church ( and are buried as Orthdox Christians if they repose prior to Baptism) are prayed for by name during the Liturgy. Orthodox Christians during the Ektanias pray for the world in general --- "those who travel by sea and by air" ," the sick and dying", "the prisoners", "the Armed Forces". national leaders ("President" "King" "Queen" etc), and the people and clergy.
As Orthodox Christians we say private daily prayers that are for anyone we wish to pray for.
1) We pray for our dead (Heterodox and Orthodox) by using the Akathist for the Repose of Those Who Have Fallen Asleep located at: http://users.sisqtel.net/williams/akathist-repose.html
2) For Orthodox Christians who have died we expand ourt prayers to include the Trisagion for the Dead located at: http://goarch.org/chapel/liturgical_texts/trisagion
The Sacraments of the Orthodox Church are composed of prayers, hymns, scripture lessons, gestures and processions---these are reserved for Orthodox Christians. "The funeral service in the Orthodox Church, although not considered as specifically sacramental, belongs among the special liturgical rites of the People of God."(The Faith, By Father Thomas Hopko)
Individually we are empowered by the Church to interface with the world through prayer, alms, fasting, and other actions. The Church on the otherhand is established primarily to serve the Kingdom of God and its Citizens (the laity and clergy of the Church). Who are the Churches citizens? Those who are catechumen (who have made the step of becoming a catechumen and are learning to be Orthodox Christians) and those who have become citizens of the Kingdom through the Sacraments of Baptism and Chrismation. The Church serves those who have made these commitments from birth thru death and continues to pray for its citizens even after death.
I hope this explains it in some small manner.