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It seems that a vapor barrier of some sort would have been installed to address that potentiality. If I were you, I'd contact Austin Patterson Disston Arcitects to find out their approach to this design. Depending on the answer, Donna Thomas may be right!

There's also the question of what climates are involved. Where's the house that has that tub? Where are you? Some places have a lot more issues with humidity than others (and how that affects condensation). There are a lot of different vapor barrier products out there, including kinds that paint-on. And potentially things like spray-on (professionally done) insulation products. But given the weight of an iron tub and then the added weight of it filled with water and people I'd be more concerned about the long term issues with the tiling joints where it meets the tub.

again, if you find a good contractor, there wont be any climate issues- i used to sell marble, granite and tile in the hamptons. there is nothing that cant be done and done right. period !!!! do it the way you want it the first time and it will last you yoor lifetime !!! dont look for the cheapest- look for the best !!! hugs :-)

Your contractor has the right idea but I don't really see why this cast iron bath tub situation would be any different than any other cast iron tub (the standard type) that is built in against an exterior wall. A typical bathtub has an inaccessible pocket of space under and around it, unlike a clawfoot tub, and that doesn't seem to be a serious problem...not that I've ever taken a look under one.

Some of the other comments here don't quite make sense to me. The purpose of the vapor barrier is to keep humid air from infiltrating the wall from the INTERIOR, believe it or not. To reduce the potential for condensation it might advantageous to install extra insulation at the exterior wall in the enclosed pocket of space under the tub. If you keep the cold out you can reduce the potential for condensation on the underside of the tub (of course cold air has a lower dew point.)

I designed a bathroom, sort of like this, for my own house (with an acrylic tub) and the tile/marble installer used this fabulous liquid rubberized waterproofing around the drain and under the marble deck (remember marble is porous...if you intend to use marble as shown in this picture) and that really reassured me about the potential for leaks under the bath/shower. Do make sure that the GC and/or tile installer have a good technical sense of how to solve these waterproofing problems.

...my bathroom below. (Sold the house just before the real esate crash, thankfully...but I do miss it!

absolutely love the tub installation!!!! i have a small mbr that i'm re-modeling, and want a large soaking tub in a small space...just beautiful.

i intend to ask my contractor about it :)

You should not have any issues with condensation. If you are concerned, I would air seal all of the framing by caulking the base plates of the walls, spray foam the exterior wall around the tub with 3" thick closed cell spray foam and 1" thick at the two side walls below the tub deck, and use Schluter/Ditra panels across the face of the tub supported by wood blocking under the tub deck. The spray foam and caulking below the tub deck will prevent air infiltration and condensation. The Schluter/Ditra systems will provide you with an uncoupling/anti-fracture membrane to prevent any cracks in the tile and it can be scribed more easily to the curved front of the tub. Most good tile installers know how to use this system. Best of luck!
Jarrett