I just airbrushed an iced cake (crusting buttercream) and realized the filling should be refrigerated. Will refrigeration cause the ink to run???

Quite possibly  Your best bet is to put it in a box and wrap it very well with plastic wrap, so that the moisture will not bead on it when you take it out. It should warm up for at least an hour or two, before you take the wrap off, and then you can do touch ups.

Hopefully it was not an intricate design and the while tier was sprayed??

Please post before and after refrigeration pics, so we all can learn  

It was just a simple sheet cake. The edges were airbrushed gold. I refrigerated it overnight and there wasn't a problem. I think I would've been really worried had it been a fondant cake. Thanks for your help!:)

I refrigerate all my cakes, even the ones I airbrush.  I have never had a problem.