We have a large Orthodox presence in Atlanta, both Greek and several different groups of Russians, so it's not a problem here, but if somebody lived in a very isolated area where no Orthodox priest was available, and some emergency came up, perhaps they would have been given permission ahead of time?
The invitation is so hedged about with cautions, it makes it pretty clear.
In my parish, there is a Greek Orthodox family that occasionally attends our Divine Liturgy. The wife is from Jordan and is drawn to the Syriac, Aramaic and Arabic texts. The husband is a convert from the Episcopal Church. There is a vibrant Greek Orthodox community nearby. They follow the guidelines in our missalettes which follow those published by the USCCB.
The guidelines for receiving Communion, which are issued by the U.S. bishops and published in many missalettes, explain, "We welcome our fellow Christians to this celebration of the Eucharist as our brothers and sisters. We pray that our common baptism and the action of the Holy Spirit in this Eucharist will draw us closer to one another and begin to dispel the sad divisions which separate us. We pray that these will lessen and finally disappear, in keeping with Christ’s prayer for us ‘that they may all be one’ (John 17:21).
"Because Catholics believe that the celebration of the Eucharist is a sign of the reality of the oneness of faith, life, and worship, members of those churches with whom we are not yet fully united are ordinarily not admitted to Communion. Eucharistic sharing in exceptional circumstances by other Christians requires permission according to the directives of the diocesan bishop and the provisions of canon law. . . . " Scripture is clear that partaking of the Eucharist is among the highest signs of Christian unity: "Because there is one bread, we who are many are one body, for we all partake of the one bread" (1 Cor. 10:17). For this reason, it is normally impossible for non-Catholic Christians to receive Holy Communion, for to do so would be to proclaim a unity to exist that, regrettably, does not.
Another reason that many non-Catholics may not ordinarily receive Communion is for their own protection, since many reject the doctrine of the Real Presence of Christ in the Eucharist. Scripture warns that it is very dangerous for one not believing in the Real Presence to receive Communion: "For any one who eats and drinks without discerning the body eats and drinks judgment upon himself. That is why many of you are weak and ill, and some have died" (1 Cor. 11:29–30).
However, there are circumstances when non-Catholics may receive Communion from a Catholic priest. This is especially the case when it comes to Eastern Orthodox Christians, who share the same faith concerning the nature of the sacraments:
"Catholic ministers may licitly administer the sacraments of penance, Eucharist and anointing of the sick to members of the oriental churches which do not have full Communion with the Catholic Church, if they ask on their own for the sacraments and are properly disposed. This holds also for members of other churches, which in the judgment of the Apostolic See are in the same condition as the oriental churches as far as these sacraments are concerned" (CIC 844 § 3). Christians in these churches should, of course, respect their own church’s guidelines regarding when it would be permissible for them to receive Communion in a Catholic church.
The circumstances in which Protestants are permitted to receive Communion are more limited, though it is still possible for them to do so under certain specifically defined circumstances.
Canon law explains the parameters: "If the danger of death is present or other grave necessity, in the judgment of the diocesan bishop or the conference of bishops, Catholic ministers may licitly administer these sacraments to other Christians who do not have full Communion with the Catholic Church, who cannot approach a minister of their own community and on their own ask for it, provided they manifest Catholic faith in these sacraments and are properly disposed" (CIC 844 § 4).
It is important to remember that, under the rubrics specified above, even in those rare circumstances when non-Catholics are able to receive Communion, the same requirements apply to them as to Catholics.