I received these potted rooted cuttings (already in this condition) yesterday. I removed all the figs, as well as any obviously dead leaves. I watered and placed it in an opaque storage bin for humidity, in the shade. Should I remove all the leaves? Any other life support advice?

Looks like a couple of my newly removed air layers. I am assuming this is a rooted cutting? If so, there isn't much else I know that you can do, just give it a couple days in the extra humidity of the bin and it should recover, then gradually open the bin and acclimate. I haven't lost any of my airlayers even when they collapsed like your plant. Good luck.

Barry,
I believe so, it came from the friend of a relative, who had all the details (written, I didn't speak with her) of how her grandfather and the Saitta family emigrated to Brooklyn, from Santa Margherita di Belice, Sicily in 1898, and brought the cuttings with them. Cuttings have been distributed through the family since then. No further details yet.
I am hesitant to disturb the cuttings any further.

The other thing- If it was an air-layer, it may have been removed too fast before enough roots formed to support it on its own.

At this point shade and humidity may be your best bet to revive it.

It is what it is, at this point, and you have done everything correctly. Now be patient and let things take their course. Likely it will shed all the leaves, but if there were roots, it will eventually stabilize and begin to grow again.

I had rooted plants back at the end of may that didn't start to grow until about half of july when i put them outside, not they are really going so you may need to wait some time, but don't give up.

It helps to mist the leaves a couple times a day in addition to watering,shade, and being in a humid container.They may rehydrate if they haven't been in a wilted condition for too long.

Awww, how sad. Take it indoors, put it in a huge translucent plastic bin, mist it every now and then. I fed mine with "super thrive" it bounced back in five days... 

Or... Let nature take its course like everyone said :-)

If it is too hot outside ,(over 90),the plant will suffer more and possibly die.
I suggest place it in air condition room ,for a few days so roots can stay alive and start feed the trunk.
Once you see new leaves starting growing,you take it out in shade.
Jennifer suggestion is complete,so you can follow her example.
If you do not have superthrive that is ok,do not feed with nothing,because at this point a poor soil works better for ,forming roots,than rich soil.

Okay, relocated to the dining room table, a balmy 74 F.
Jennifer, it's in the biggest storage container I have- 12 gallons or so.

Thank you everyone for the feedback!