I studied purgatory and despite hearing some Orthodox Christians say otherwise, I came to the conclusion that the Orthodox church and most if not all Orthodox Christians believe in purgatory as Rome defines it. If there was a horos or universally received council that discussed purgatory in a clear and precise way like the first seven ecumenical councils then I would know for sure whether the Orthodox church rejected Rome's teaching on purgatory but I'm guessing there wasn't a council that discussed it (other than the Council of Florence where Orthodox representatives accepted Rome's teaching on purgatory).
Rome believes there are two fires. A temporal fire used for purgation where the soul experiences pains of various durations and intensities till it is cleansed of certain sins. Prayers of the dead can speed up the cleansing process but their salvation is assured regardless. They also believe in a second eternal fire of Gehenna reserved for the wicked after the second coming.
Orthodoxy rejects any notion of a temporal fire of torture as the forgiveness mechanism God has chosen to cleanse away sins sins. We believe that forgiveness and punishment are mutually exclusive and cannot be used simultaneously. Foregiveness is freely letting go, punishment is vengeance to appease, thus either God will forgive your sins or He will punish but not both as these two things are polar opposites. Secondly Orthodoxy rejects tortures upon the soul apart from the body whose passions are tied to these sins, thus there can only be one eternal fire after the ressurection when the soul reunites with its body.
Orthodoxy also believes you receive only a foretaste of what is to come not the fullness of bliss or eternal seperation, so there is no need for a temporal fire to 'graduate' into salvation. We have not been perfected yet nor damned without hope
In the Roman Catholic doctrine, prayers for the dead are exclusively for those in purgatory (or in hell) that will eventually aid in their release unto heavenly salvation. (If they are punished in the purgatorial fire it doesnt really matter as they are assured of release, so its a moot point.)
In Orthodoxy, prayers for the dead are not understood in such a way. Prayers for the dead can bring relief to souls imprisoned but does not guarantee their release, in other instances prayers for the dead can improve the situation of a sinful soul as the gospel teaches:
Anyone who speaks a word against the Son of Man will be forgiven, but anyone who speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come. Matt 12.32) Thus all kinds of sins will be forgiven in the age to come (except those against the Holy Spirit) but not through a torturous temporal fire.
But heres the caveat, In Orthodoxy prayers for the dead really have nothing to do with aiding of sinners and releasing one from hell etc. For Rome this is the primary factor for prayers of the dead, so an innovation like purgatory fits into their self- understanding of whats on the other side of the grave, but not so in Orthodoxy.
In Orthodoxy prayers for the dead is for the remembrance of our loved ones who have died in the faith. Hence the holier the person is the more prayers he will be commemorated with. Its based on the love of the deceased because Love alone endures forever. This is why i know of family members who commemorate deceased loved ones that have died over 50 years ago and make kolyva (wheat) for them as an offering to the church and congregation. Because this tradition from time immemorable echos what Paul taught:
..'But someone will ask, “How are the dead raised? With what kind of body will they come?” 36How foolish! What you sow does not come to life unless it dies. 37When you sow, you do not plant the body that will be, but just a seed, perhaps of wheat or of something else... 42So will it be with the resurrection of the dead. The body that is sown is perishable, it is raised imperishable; 43it is sown in dishonor, it is raised in glory; it is sown in weakness, it is raised in power; 44it is sown a natural body, it is raised a spiritual body. 1Cor 15.35,42)
When my family and I pray for my deceased grandparents and relatives, its not because we think they are in hell or in some purgatory but because they are members of the Church triumphant, whose perfection is not yet complete, thats why the church offers prayers on our behalf on two different continents.