What will happen if Arab Muslims pray inside a synagogue?

Nothing. They can pray in a synagogue if they want but by “pray” I mean what Muslims call “du’a” not really what they call “salah”.

A synagogue is not a fit place for Muslims to perform their ritual prayers. Because we wear our shoes in the synagogue, the floor is not git for prostration and there is not much room to do it anyway. Synagogues have pews because we pray standing and sitting but not orostrating on the floor- a practice that was done in the Temple. Actually, we do orostratebourselves once a year on Yom Kippur but otherwise not.

So, Muslims could certainly make du’aa- or non-ritual prayer, in a synagogue but they would probably not find it acceptable as a place to use as a masjid and would not really want to do it there.

Similarly, Jews may pray in a mosque (obviously if Muslims give permission) because mosques have no idols nor anything else forbidden to Jews in them and they are considered “spiritually clean” spaces. İn practice, however, most Jews would not pray in a mosque because we wouldn’t want to disrupt the Muslim worshippers with our presence. Nonetheless, there is nothing forbidden about praying in a mosque if Muslims invite us to do so and there have been a few cases where Muslims have allowed Jews with no synagogue to temporarily use a mosque to hold Jewish prayers ( at separate times to the Muslim prayers).

These kinds of intercommunal kindnesses and outreach are wonderful when they happen but, for many reasons, they don’t happen that often.

Some synagogues may host Muslim guests for certain events and they may be invited to say a prayer while there. The late great Muhammed Ali (may his soul be blessed in Heaven) would sometimes be hosted at the synagogue of a Rabbi I am acquainted with and would often be invited to say a prayer. So, certainly it happens.