Can the paper-corded seats on Moller chairs be cleaned?  

Hmm.
Interesting cleaning problem.
An artist's kneaded eraser? A 'Mr. Clean Magic Eraser' perhaps?
I'd research paper-cleaning techniques to try to come up with the solution.
http://www.carrmclean.ca/CategoryGroupBrowser.aspx?CategoryID=233&GroupN...

I doubt it, unless they...
I doubt it, unless they finished it somehow.
Have you ever "cleaned" paper?

Soap Flakes
The Moller seats are the same type of papercord as Wegner seats. Use soap flakes in water, wipe it on, let it dry. Repeat every couple of weeks.
I was able to clean up a Wegner rocking chair seat this way. I also lifted most of a coffee stain out of another paper-cord seat.
Recipe on soap-flakes.com below.
http://www.soap-flakes.com/recipes.html

Yes, I've cleaned paper
thousands of times, usually by means of an eraser; kneaded-artists erasers being the most effective.
I've removed gooey sticker residue from paper (books) using lighter fluid.
Perhaps some sort of dry-cleaning solvent would work for paper cord?
These are just guesses, of course; try them at your own risk.
This cleaning-problem captured my imagination because I had a set of these paper-corded chairs with a stray food stain or two. I always meant to tackle those stains, but sold them before I had the chance.

Lucifersum to the rescue!
Soap flakes, eh? I'll have to remember that.
As with any DIY solution, the results will vary, depending upon the finesse of the DIYer. (I imagine that patience & delicacy is key, when applying moisture to paper cord.)

like the soap flake hint.
You could also drop an e-mail to Richard, the WickerGuru...
He is very generous with his knowledge.
His restoration videos here, http://www.youtube.com/wickerguru
And a direct link to the website here,
http://wickerworks.weebly.com/cane-raw-materials.html