If I apply royal icing to a fondant covered cake can I stick it in the fridge? How about if I apply royal icing to a buttercream covered cake?

Yes, on fondant -- never tried royal on buttercream. I use buttercream on buttercream and then, yes, fridge is fine.

Thank you luddroth.

I'd be concerned about condensation when it comes out of the fridge. Moisture can break down royal icing. With the weather cooling off it may not be that big of an issue for you, but I'd be careful myself.

TexasSugar is right -- but the same thing is true of the fondant itself. Whenever fondant/royal has been in the fridge, it should be covered as it comes out if the weather is humid -- condensation is an enemy of both fondant and royal icing.