The result is drying out insulation etc reducing weight, but also transferring that moisture to the cabin air.
If it indeed just transfers moisture from an area where there is an excess of it to an area where it is insufficient (for passenger comfort), both areas being part of the same plane, then the net balance will be zero and no fuel will be saved. If anything, some extra fuel will be burned in order to power the units.
Edit to add: and the units themselves have to be flown around, as pointed out by catttletruck.