Why Pray For the Dead?

When you die you will face what is known as the partial judgment. This will include a complete examination of your life.  With a good account you will be led by angels to a mystical place where we will anticipate the joys of Paradise awaiting the final judgment and your resurrection.  If you do not know God at this point and have not lead a life of repentance you will be controlled by the demons who will lead you to a place where you anticipate the torments of Hell or an eternal life separated from God.  
Elder Cleopa tells us this about those who are destined for eternal torments,

"If someone at the partial judgment is destined for eternal torments and is a Christian and servant of Christ, he has but one hope.  His hope is in the intercession of living Christians who are able to pray to Christ for him to be rescued from the torments of hell or at least to find some relief from them."

Paul tells us, "we must appear before the judgment seat of Christ, that every one may receive the things done in the body, according to what he hath done, whether it be good or bad." (2 Cor 5:10)  but Elder Cleopa points out that Paul also said to Timothy, "I exhort therefore, first of all, that supplications, prayers, intercessions and giving of thanks be made for all men." (1 Tim 2:1)  Also James says, "Confess your sins to one another, and pray one for another, that ye may be healed. The effectual fervent prayer of a righteous man availeth much." (James 5:16)

Elder Cleopa says,

Consequently, if our prayers are able to benefit the living for what reason are they powerless to benefit the dead, granted that they also live by their souls? God is everywhere present and hears both the prayers for the living and for the dead.

We can see in the Old Testament witness to prayers for the dead (2 Mac 12:442-45, Bar 3:4-5).  We also see in Holy Tradition and in the Divine Liturgy prayers for the dead.

Praying for the dead does not place our hope of salvation in the hands of humans. Those who are separated from God will not be saved, but those who have their hope additionally in the prayers of men of faith may be helped through their prayers much like Paul depended on the prayers of his followers.  We must remember that God is all powerful with unlimited goodness. He is surely able to rescind the eternal anguish of man.  He asks for our love and our love of each other.  When we pray for each other this is an act of love. We know the Theotokos and the angels and all the saints are always praying for us especially when we join with them in our services, such as a memorial for the dead or the Divine Liturgy.  Jesus told us, "Therefore I say unto you, what things soever ye desire when ye pray, believe that ye receive them, and ye shall have them." (Mk 11:24)  Elder Cleopa says, "Consequently, prayer for the reposed is not only a sign and strengthening of the love we share between us, but also proof of our faith. Thus the Savior says, "If thou canst believe, all things are possible to him that believeth." (Mk 9:23)

In the parable of the rich man and Lazarus (Luke 16) we are told of a great chasm that exist between heaven and hell.  
Elder Cleopa tells us, 

"Yet, this chasm does not have the power to impede the mercy of our great God, Who hears our prayers for the reposed. We do not suppose, as do the Roman Catholics that there exists a purgatorial fire, but we say that only for those who sinned very severely and did not confess their sin is the passage from Hades to Paradise impossible.  For those who sinned more lightly this pathway is not definitely closed, given that in the future judgment each one's place, either in heaven or hell, will be decided definitely, inasmuch as after his judgment someone whose orientation was Hades can no longer pass over into Paradise. For those who sinned unto death, our prayers are completely futile...

God looks down from the heavens with attentiveness upon that which springs from love, for love is in its entirely the sum of His commandments."

Reference: The Truth of our Faith, 123-133