According to Catholicism or Eastern Orthodoxy are there any doctrinal barriers to Christians sharing places of worship with non-Christians?
According to Catholicism, it theoretically seems to be possible, but only as an emergency situation, if it is not a consecrated church or cathedral. Please remember that Catholic churches and cathedrals are consecrated to the service of God and all danger of of a sacrilege and scandal must absolutely be avoided. Churches in France, were used as military hospitals during World War I, but the Blessed Sacrament was removed from the churches in question.
As far as Catholicism, goes there seems so be very little written on this subject, but Canon Law seems to forbid it. Muslims are not Christians and their acts of worship in a Catholic place of worship would constitute a profane act in itself. It would be very close to an act of desecration, seeing that such sacred places house the Blessed Sacrament.
The late Pope St. John Paul II gave a Catholic church to one of the Orthodox Churches in Rome for their own usage when they could not acquire one for themselves. But then the Orthodox churches have the faith in the same seven sacraments of Catholics.
One thing to keep in mind is that Muslims are not Christians and historically much bloodshed has been spilt by both sides. Sharing a ”building” may in modern times may be permitted, due to some extenuating circumstances, but in no way should this be a permanent situation for fear that tensions may build up nor should the building be a consecrated church. A rented hall could be a viable solution, but not a sacred space of worship such as a church.
As far as the Catholic Church is concerned, Muslims worshipping in a sacred space such as a church would be an act of profane usage and is forbidden.
Here is what Canon Law has to say about Sacred Places:
1205 Sacred places are those which are designated for divine worship or for the burial of the faithful by a dedication or a blessing which the liturgical books prescribe for this purpose.
1210 Only those things which serve the exercise or promotion of worship, piety, or religion are permitted in a sacred place; anything not consonant with the holiness of the place is forbidden. In an individual case, however, the ordinary can permit other uses which are not contrary to the holiness of the place.
1220 §1. All those responsible are to take care that in churches such cleanliness and beauty are preserved as befit a house of God and that whatever is inappropriate to the holiness of the place is excluded.
1222 §1. If a church cannot be used in any way for divine worship and there is no possibility of repairing it, the diocesan bishop can relegate it to profane but not sordid use.
§2. Where other grave causes suggest that a church no longer be used for divine worship, the diocesan bishop, after having heard the presbyteral council, can relegate it to profane but not sordid use, with the consent of those who legitimately claim rights for themselves in the church and provided that the good of souls suffers no detriment thereby.
Catholics and Orthodox may not allow this in their churches, but the Lutherans seem to be a home with the idea.
In fact, a Lutheran Church in Berlin has opened its’ doors to Muslims due to the Covid-19 we are presently experiencing.
A church in Berlin has opened its doors to Muslim worshippers unable to fit into their mosque under new social distancing rules.
Germany allowed religious services to resume on 4 May but worshippers must maintain a distance of 1.5m (5ft).
As a result the Dar Assalam mosque in the city's Neukölln district could only hold a fraction of its congregation.
But the Martha Lutheran church in Kreuzberg offered to help by hosting Friday prayers at the end of Ramadan.
Throughout the month of Ramadan, Muslims abstain from eating, drinking, smoking and sex from dawn to dusk. Normally families and friends would gather to break their fast and attend communal prayers, but in Berlin - as in countries across the world - this year's celebrations have been affected.
Ramadan: German church opens doors for Muslim