That's more or less what heat pump clothes dryers do. They draw in air, heat it, and then blow the humid air back over the cold side of the heat pump loop to condense it. They save tons of energy. I don't think there's anything similar for water (in clothes or dish washers) but the quantities might just be too small to bother with.
a counter current heat exchanger might be more reliable than a refrigerant heat pump and more efficient than peltier heat pumps, but yes in theory you can quasistatically move the water out for almost no energy (only the adsorption / mixing energies). And peltiers can be much more efficient at tiny temperature gradients, so a theoretical frictionless positive displacement pump, and a heat pipe between the condensing compartment and the evaporatig compartment (so keeping the 2 compartments at ambient temperature (so no heat losses).
Compressing humid air releases heat which marginally increases the temperature of the condensing compartment, and marginally cools the evaporating compartment with the wet clothes; thus the heat pipe will quickly equilibrate the 2 compartments.
