I'm not quite sure what the problem is with praying in a place of worship of the wrong denomination - as long as no idolatry is involved - but generally, Jews do not pray in Mosques, Muslims don't pray in synagogues, Jews don't pray in churches, or vice versa, though churches might also have idolatry issues, depending on the exact nuance of religion...
Ladaat is reporting on a Muslim who went to do his prayers in the synagogue of Ben Gurion airport. He was discovered when a minyan of Jews went in to daven the evening services.
I don't know if religiously it is a big deal or not. It does not seem to be such a big deal... except for one thing... the Muslims flip out when Jews might even potentially pray in their mosques. See how they act on Har Habayit for example, with their complaint being that the entire area is considered part of their mosque.
I would say that with them being so uptight about it, they should not just go ahead and use another religion's prayer house at their whim.
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