I've iced many birthday cakes the day before and put it in the fridge and never had a problem with it. Firstly, take it out an hour or so before its time to eat it so its not too cold, but generally the butter icing acts as a barrier so the cake doesn't dry out the same as if it was put in the fridge without anything on it at all.
I generally find for some reason the icing seems brighter and a better colour after putting it in the fridge to set or keep.

Having said that, I've also iced a butter icing cake in the morning and left it out of the fridge til the afternoon cause there was no frdige space and it wasn't a particually hot day and it was fine aswell.

I generally do mine ahead of time and refridgerate just to be organised and save doing it last minute.

I've been making and decorating cakes for years and years and one in particular (3D Elephant cake) took 5 hours to decorate and i would do some, stick it in the fridge, take it out, ice some more etc

Good luck with the cake!