Many years ago the insulation was removed from a 767 during maintenance and when it was reweighed well over half a ton of moisture had been removed.
To me that says the problem is with the insulation material that is used in that it also has the additional attributes of a sponge. I imagine it would be doing a poor job insulating if it were wet or iced up, which would put even more pressure on cabin heating.
A better solution than installing moisture extraction fans would have been to using insulating material that does not absorb or is impervious to condensation in the first place.