Its not like the person that owns the building changes because they lease it out.
Right, but as with our Art Deco Museum which also is a non-profit, if we make income with something that is not related to our mission, then we must pay taxes on it. We pay taxes on items sold at our gift shop for instance, but donations to the museum that are used to further the mission of the museum, are not taxed.
I am guessing that if we had our own stand alone museum and part of the building was for gallery/office space and part of the building was a gift shop and restaurant, the portion that was not office/gallery space would be taxed (sure there are caveats like, if the amount of space were over such and such and we made more than X amount, etc.). It may even be that part of the Philcade that is being used for the museum may be able to get some tax deductions and such, but really have no idea.