As an Anglican I am sure that there are no qualms about "performing" liturgical texts outside the church. I think the Roman Catholic position is the same. (There is a tradition of not singing Glorias or Alleluias during Lent, though, in or out of the church).
But I have heard that there are some Eastern Orthodox liturgical hymns which, in their tradition, are prescribed to be used only in the church and only at certain times at that. Any 'catters who can enlighten us here? And going even farther afield from my "known world," I understand there are indigenous songs/chants/whatevers which the followers of those tradition feel quite strictly are profaned by singing them out of place.
So the original question was a good one. We do want to be at least a little careful about not offending those to whom these things may be very sacred.
-HM