On this topic, it might be helpful to read the SCOBA-promulgated "Guidelines For Orthodox Christians in Ecumenical Relations," which spells out the principles and specifics for all such things, both on an official level (e.g. ecumenical prayer services) and on a more local level (e.g. how should a priest conduct an Orthodox wedding if one of the members is not Orthodox; or in what way can he attend such a couple's wedding at a non-Orthodox church). Of course, this document is only representative of the canonical Churches in the Americas, but it does give one some idea of what might happen at the Oecumenical Patriarchate.
The full document is available here:http://www.scoba.us/resources/documents/guide_for_orthodox.pdf
I have pasted the most relevant sections below:WORSHIP WITH NON-ORTHODOX
1. The Orthodox Church makes a clear distinction between liturgical and non-liturgical prayer. Our liturgical prayer is the prayer and devotional action of the holy Orthodox Church. In this sense, liturgical prayer is the official prayer of the Orthodox Church and is to be conducted according to the forms, prescriptions and canons of our Church.
2. Non-liturgical prayer can be understood in two senses. In one sense, it is the private prayer or devotions of the faithful Orthodox. In this sense, it is also the prayer of the Church and has ecclesial character and significance, insofar as one prays within the context and life of the Church. In another sense, however, non-liturgical prayer may be understood as that private or corporate prayer of divided Christians from diverse communions who come together, not as the Church, but as separated brethren seeking Christian unity. It is common prayer of non-ecclesial character. It is to be prudently used within the context of the ecumenical movement and the pluralistic setting of our society.
3. The basic Orthodox conviction has always been that unity at the altar, the unity of the members of the Orthodox Church, is a gift of God. The celebration of the Holy Eucharist and the reception of Holy Communion in the Divine Liturgy is the final end and goal of the Christian life, the very fulfillment of unity. All of the services and prayers of the Church are intimately connected with the Liturgy and express that gift of unity which is given to us by God and preserved in the bonds of faith and love. Therefore, the services of the Church are restricted to the members of the Orthodox Church and must not be understood or implemented as means toward that unity. As a sacramental community of faith and grace, the Orthodox Church in its self-understanding and with full responsibility for the Apostolic Faith which has been entrusted to it, encourages liturgical worship and frequent participation in the sacraments for its own members. At the same time, it encourages non-liturgical prayer for the union of all people, for peace, reconciliation and the spirit of charity.
4. In the interests of sharing our spiritual heritage, non-Orthodox may be invited to attend Orthodox liturgical services. It should be made absolutely clear, however, that no communicato in sacris
is intended or implied by such attendance. The same is true for those Orthodox who for reasons of family unity, courtesy, the demands of public life, or a deeper appreciation of the worship of other communions might be invited to attend a non-Orthodox denominational service. In extending or accepting such invitations, care should be taken not to offend against the regulations or sensitivities of other communions.
5. Clergy of other communions attending Orthodox services may be welcome as guests of honor, and given some special place within the solea. High dignitaries of other Churches, when the formal occasion indicates, might be seated adjacent to the BishopÃ¢â‚¬â„¢s throne when a Bishop is present. Civic authorities may be seated in the first rows or opposite the BishopÃ¢â‚¬â„¢s throne on the solea.SPECIAL COMMON PRAYER PRACTICES
1. A clergyman is free to accept invitations to observances of a civic, patriotic or general community nature. If invited to offer a prayer at such an observance, e.g., school commencement, Independence Day, Memorial Day, Veterans Day, banquets, United Nations observance, etc., conducted in a public place or a neutral hall, the participating priest may accept the invitation but should not wear any form of liturgical vestment. The prayer should be composed for the occasion by the priest, reflecting the Orthodox attitude toward the issue as found in our Service books, but also respecting the spiritual sensibilities of all the participants who are inevitably of diverse backgrounds.
2. In services of an interfaith or interreligious nature, e.g., national feasts, public calamity and mourning, Brotherhood Week, the dignity of the family, expressions for peace, justice and the like, whether in a public building or a religious edifice, a form of dress which is neither liturgical nor merely civil, viz. the rasson (cassock) may be considered appropriate, together with pectoral cross (if so entitled), or academic dress when indicated. No part of the liturgical vestments, such as stole, is proper.
3. Ã¢â‚¬Å“Ecumenical servicesÃ¢â‚¬Â refer to forms of non-liturgical worship or devotion mutually acceptable to all participating parties in which Christians of various communions take part. Although such services are concerned particularly with the restoration of Christian unity they may be held for any common concern in which Christians can and should cooperate with one another.
4. Ã¢â‚¬Å“Ecumenical servicesÃ¢â‚¬Â may be conducted in an Orthodox Church with the permission of the Bishop. Furthermore, Orthodox Christians may take part in such services in the churches of other communions, as well as in other appropriate locations.
a) In order to avoid any misunderstanding, however, these services should be publically acknowledged and identified as Ã¢â‚¬Å“ecumenicalÃ¢â‚¬Â in character, emphasizing the firm Orthodox position that these are prayers for unity, and not services of the one Church.
b) An Orthodox priest should not wear liturgical vestments at such services. The rasson and pectoral cross (if so entitled), or academic dress are appropriate.
c) If invited to participate, the Orthodox priest should share in the preparation and planning for an Ã¢â‚¬Å“ecumenical serviceÃ¢â‚¬Â and contribute to its proper form and content. Prayers and petitions from the Orthodox Service Books are recommended.
d) When Ã¢â‚¬Å“ecumenical servicesÃ¢â‚¬Â are conducted in an Orthodox Church, the host Pastor may compose an appropriate service based on the prayers and forms of the Orthodox Service books. Clergy and laity of other communions may be invited to read the Scriptures, offer prayers and give invocations. Clergy of other communions may be invited to preach. All such services in an Orthodox church must take place outside the ikonostasion, in the area of the solea. Occasions for Ã¢â‚¬Å“ecumenical servicesÃ¢â‚¬Â are usually provided during the Week of Prayer for Christian Unity (January 18 25); the days from Ascension to Pentecost; and on the occasion of meetings or other events of ecumenical origin serving on ecumenical purpose. Although petitions and prayers for unity are a regular part of the Orthodox liturgical practice, Ã¢â‚¬Å“ecumenical servicesÃ¢â‚¬Â may be encouraged as a means of sensitizing our faithful to the tragedy of Christian disunity and developing the spirit of charity, understanding and prayer for all persons.
e) Orthodox clergy and laity are free to read Scriptures, offer prayers, and give invocations at Ã¢â‚¬Å“ecumenical services.Ã¢â‚¬Â Likewise, Orthodox clergy may preach on these occasions.