I made a cake with a filling that must be refrigerated. If I cover it in MMF and refrigerate the cake overnight will that hurt the MMF?

No, it won't hurt it, and with a filling that has to be refrigerated, you don't have any choice. It may develop condensation when you take it out of the fridge, but don't touch it while it's wet and the condensation will evaporate.

no, I just made a cake last night with MMF and refrigerated it all night and it looked exactly the same.

My concern was that it may get hard. I've only worked with MMF one other time and didn't have to refrigerate it.

I have refrigerated it before and it acted the same as regular fondant, it did not get hard.

I refrigerate all of my fondant cakes overnight, and they do great. Just know when it comes out, condensation will form. Just don't touch it until the condensation evaporates, or it will leave a fingerprint. Once the condensation evaporates, it looks exactly the same. I have never had any issues refrigerating my MMF.