I’ve kept cakes for up to a week with no ill effects. (I live in North Carolina where we do hot and humid pretty intensely in summer, though not quite of the tropical variety.) Specifically, I’ve kept an unrefrigerated cake covered in ganache & fondant, uncut, for three days. Once cut it was fresh for a couple of more days before it started showing signs of staleness (though my teenage son didn’t mind). The cake was a very stable one, not too moist, and the fillings were all shelf-stable buttercream. I’ve also kept a fondant covered cake (no ganache) with buttercream filling for a day before cutting, and it stayed fresh another three or four days. Also, I do cover the cut face of the cake with plastic wrap (the part that shows the layers) but not the fondant part.
I never refrigerate my fondant cakes; the condensation makes the fondant sag and leaves the surface … funny-looking. I now have a dehumidifier and I am hoping to try to refrigerate a cake and place it upcloseandpersonal to the machine to see if it dries quickly enough to prevent marring the surface.