The two headed eagle flag in the Protaton Church in the capital of Mount Athos Karyes
The yellow eagle flag on the Esfigmenou monastery with the black flag which represents the resistance of the monks against Patriarch Bartholomew who stands for a better relationship with the Vatican.
The Orthodox church of the exaltation of the cross at Perissa, Santorini where I stayed with my family in may this year.
The doubleheaded eagle flag is not only a symbol of the orthodox church, This for instance is the flag of the village of Naarden in the Netherlands showed by your webmaster. A universal symbol but mostly the symbol of the Orthodox church.
hv
Athos - Agios Oros 
“The double-headed eagle is a common symbol in heraldry and vexillology. It is most commonly associated with the Byzantine Empire, the Holy Roman Empire, the Russian Empire and their successor states. In Byzantine heraldry, the heads represent the Emperor having authority over both secular and religious matters, Byzantine emperors were regarded as Christ’s vicegerent on Earth. It also signified the dominance of the Byzantine Emperors over both East and West. In the Holy Roman Empire’s heraldry, it represented the church and the state. Several Eastern European nations adopted it from the Byzantines and continue to use it as their national symbol to this day.”
http://en.wikipedia.org/wiki/Double-headed_eagle
In the West it’s seen as the last flag of the Roman Empire, not necessarily Orthodox. It was also used by the Roman Catholic Holy Roman Empire.