Is there anything in Jewish law restricting Jews from working for other religious organizations? Some examples: can a Jew work as a receptionist at a Mosque? As a gym teacher at a Catholic school? As a professor at a Mormon university? etc.
No, (it may not be considered proper by some, but) there is no such specific restriction.
There are several cases in the Tanach in which even "Gdolei Hador" have worked in various places, from Yaacov at Lavan's, Yosef at Pharaoh's palace and onward...
When talking about a Mosque or a Church, one should know that there is a din restricting entrance to a place of foreign worship (See Gmara, Avoida Zara page 11b) and since for example christianity is Idolatry, it is forbidden to enter a Church. The Rambam states (Mishna, Avoida Zara, Ch. 1, Mi. 4) that you are almost not allowed to even look at it.
Regarding a Mosque - there is a debate among the poiskim if it is included in that definition, so for some poiskim it may be that one can enter a Mosque, and perhaps even work there in case it is a necessity.
JPS Tanakh 1917:
And God spoke all these words, saying:
I am the LORD thy God, who brought thee out of the land of Egypt, out of the house of bondage. Thou shalt have no other gods before Me.
For Jews specifically, this should be the number one thought. It is so important Daniel refused the king's meat, and the 3 young men refused to bow to the statue of Nebekenezzer.
That's the first part of the foundation of my argument. When you place yourself under the authority of someone who serves another god, you are submitting to that authority. As potential evidence I give you Job. He was tested by the adversary. In that testing all he had was touched and destroyed. That included his children, his livestock, his home, and his servants. His employees.
When I worked for a Christian, he refused to work on Sunday because it was his holy day. By not working, I passively was honoring that day with him. This is an example I am using to make a point. If you work for a Catholic charity, you will be under the authority of an organization that practices Idolatry, claims the Jews have lost their place as the Children of God, have claimed they can change the word of God, the seasons of God and the Laws of God and hold up the Sunday command as proof of this.
They have consistently been against your people in every single point since the exile. They were at Spain, they crowned the emperor of France, Constantine, and all who followed answered to Rome. Even now Rome sides with your enemies. Why would you want to work for someone who considered their Pope the Vicar of God, the only voice between Man and God?
For now that's my answer. I hope it answers it. It is not good to serve anyone, unless you are willing to leave the moment they demand you bow to their god. and tyat means the moment they expect you to do what their god demands or expects. God is not God only one day a week, or only during prayers. He is God always. And if you need a reference for that, I have no way to help you understand better. Ask God.