Are there any biblical examples of Christians in the Early Church communicating with dead saints in order to request their intercessory prayer?
The short answer seems to be no.
There are no clear examples in Scripture (from the Book of Acts or Revelation) from the day of Pentecost or afterwards. The Day of Pentecost marks the Birthday of the Early Church! Even prayers to Jesus didn't start until after He had ascended to the Father, so those praying to him did not believe him to be dead. Nor do Catholics believe the saints are dead!!!
Only their earthly bodies cease to live. They are very much alive in spirit, while in Heaven.
Before going on, I would like to object to the phrasing of the question. In saying that some Christians, particularly Catholics communicate with dead saints in order to request their intercessory prayer, makes Catholics seem like their practice is a dark form of spiritualism or séance. Nothing is further from the truth.
The intercession of the saints is a totally different matter. The intercession of the saints is a doctrine held by the Eastern Orthodox, Oriental Orthodox, and Roman Catholic Churches.
We simply pray to our favourite saints in Heaven or for the souls of our family members that have passed away each day; yet to hold a special conversations with any of them. We do not preform séances.
We pray to the saints and we trust that our prayers are heard. Do not know why you used the phrase ”communicate” the way you did, but so be it.
As for praying to dead saints goes that is not how we see the subject matter. The saints to us are very much alive and not dead. They are very much alive in Heaven or in purgatory awaiting Heaven!
Thus Catholics pray to and asking for help from saints that are very much alive. We pray to Jesus and the saints. For us prayer is a means of remaining in communion with those in Heaven or purgatory. We worship only the Divine Trinity. The communion of saints is not an either or issue, it is a family affair!
This question gets more complicated if one takes into consideration that what applies to the Saints in Heaven also applies to the Holy Angels in Heaven.
We see in the Sacred Scriptures that the Holy Angels aid us in are prayers to God! We see this in the Book of Tobit:
12 Tobit, when you and Sarah prayed to the Lord, I was the one who brought your prayers into his glorious presence. I did the same thing each time you buried the dead. 13 On the day you got up from the table without eating your meal in order to bury that corpse, God sent me to test you. 14 But he also sent me to cure you and to rescue your daughter-in-law, Sarah, from her troubles. 15 I am Raphael, one of the seven angels who stand in the glorious presence of the Lord, ready to serve him. - Tobit 12:12-22
Yes, we can pray directly to the Father. Yet St. John’s imagery in the Book of the Apocalypse sees the four living creature and the 24 elders presents the prayers of the saints to God.
8 And when he had taken it, the four living creatures and the twenty-four elders fell down before the Lamb. Each one had a harp and they were holding golden bowls full of incense, which are the prayers of God’s people. - Revelation5:8
We see in 2 Maccabees that the inspired authors tells us of the goodness of praying for the dead. Protestants however do not include this Maccabees in their biblical canon, but it has been in our biblical canon since 382 and is recognized by Catholics and Orthodox alike. This also applies to the Book of Tobit.
46 It is therefore a holy and wholesome thought to pray for the dead, that they may be loosed from sins. - 2 Maccabees 12:46
Praying to the saints in either Heaven or purgatory is most befitting in our minds.
What is the basis of the teaching that souls in purgatory can intercede for the living?
First of all, we must remember that there are three divisions that make up the Communion of Saints: the Church Triumphant (those in heaven), the Church Suffering (those in purgatory) and the Church Militant (those among the living on earth). Yes, we are one big family in faith.
God can work miracles through his saints even though they are no longer amongst us here on earth. The Prophet Elisha brought a dead back to life. Would not the recipient of that singular gift thank both God and the Prophet Elisha in his prayers!
And it came to pass, as they were burying a man, that, behold, they spied a band of men; and they cast the man into the sepulcher of Elisha. And when the man was let down and touched the bones of Elisha, he revived and stood up on his feet. - 2 KINGS 13:21
Praying to the saints can be seen in the Early Church. The Roman catacombs bear this out quite nicely. That is not a new doctrine in Catholicism.
Do the Roman catacombs confirm the early Church believed in the doctrine of the communion of saints?
In the catacombs, sometimes rich or more well to do would leave some sort of memento at the tomb of a saint as a reminder of a particular saint to pray for them. The poor and slaves would simply write on the walls things like "Petri ora pro Victor" (Peter pray for Victor). I can still remember seeing this sort of graffiti along with various other debris in the catacombs when I was in Rome, notably la scavi. It is all too evident, that the catacombs prove that the early Christians believed in the communion of saints. Giovanni Battista de Rossi's work La Roma Sotterranae (1877) contains much information on such subjects found in the Roman catacombs.
The Church has had a long tradition that St. Peter’s Basilica, construction of which was funded by the Emperor Constantine, was built in the early fourth century atop the burial site of St. Peter. But in 1939–less than 100 years ago–a team of workmen digging a grave for Pope Pius XI in the crypt beneath the Basilica uncovered what was plainly the top of a Roman building. The new pope, Pius XII, ordered further investigation; and archeologists gradually unearthed a well preserved Roman necropolis, or city of the dead, immediately beneath the foundations of St. Peter’s.
On one graffiti wall, amid Christian symbols and petitions, the name of Peter is carved at least twenty times, usually accompanied by prayers for the dead person, and in one case expressing joy that the deceased relative lay in the same cemetery that held the body of St. Peter. - How Does the Vatican Know That Those Old Bones Are the Relics of St. Peter?
The Graffiti Wall, on the north side of the Tropaion, showing the repository. (Fabbrica di San Pietro)
Unknown to the archaeologists at the time, there had been a serious blunder early in the excavation. Early in 1942, when the team had first discovered the Graffiti Wall at the north side of the Tropaion, with its curious marble receptacle slightly exposed, they had initially chosen not to disturb it, wishing to fully photograph and document the valuable graffiti before risking damage to the plaster by reaching into the hole. When they finally did examine it, they found nothing inside but a medieval coin, some bone fragments, and other debris. - The Bones of St. Peter
The bones found beneath the Red Wall (briefly replaced there for the photograph). (Fabbrica di San Pietro)
This structure is perpendicular to the red wall, built on the right-hand side (north) of the trophy of Gaius during the second half of the 3rd century. The wall gets its name from the surprising amount of Latin graffiti carved into the plaster by the faithful who visited the tomb of St Peter between the end of the 3rd and the beginning of the 4th centuries.
The graffiti includes names, petitions, Christian symbols of complex interpretation and signs superimposed on one another on the reduced surface of the wall.
In abbreviated form, St. Peter's name is present on the wall at least twenty times, usually accompanied by prayers for the dead person named - in one case expressing joy that the lost relative lay in the same cemetery that held Peter's own body. On every part of the wall - freestanding between the letters of a name, formed from or engrafted onto existing lines - there occurred the initials PE or PET. - Necropolis (Scavi) Graffiti Wall G The bones of St Peter
As for the Early Church goes, there are numerous examples of this practice. I’ll limit myself to two Church Fathers.
Ephraim the Syrian
"You victorious martyrs who endured torments gladly for the sake of the God and Savior, you who have boldness of speech toward the Lord himself, you saints, intercede for us who are timid and sinful men, full of sloth, that the grace of Christ may come upon us, and enlighten the hearts of all of us so that we may love him" (Commentary on Mark [A.D. 370]).
"Remember me, you heirs of God, you brethren of Christ; supplicate the Savior earnestly for me, that I may be freed through Christ from him that fights against me day by day" (The Fear at the End of Life [A.D. 370]).
Cyprian of Carthage
"Let us remember one another in concord and unanimity. Let us on both sides [of death] always pray for one another. Let us relieve burdens and afflictions by mutual love, that if one of us, by the swiftness of divine condescension, shall go hence first, our love may continue in the presence of the Lord, and our prayers for our brethren and sisters not cease in the presence of the Father’s mercy" (Letters 56:5 [A.D. 253]).
Further information may be gleaned from the following articles: