What application do you have in mind?

Cellulose absent water is extremely inert, so inert that we use it as long term information storage, it is also an extremely good insulator.

You'd be more likely looking at using the cellulose as a non-conducting carrier than as part of the logic itself.

Are there an any advantages to using hydrodynamic vs. electrodynamic forces for carrying signals? Hollow cellulose nanotube structures could be printed to carry fluid to different areas of a three dimensional circuit.

Would you be able to sacrifice some of the raw "speed" you gain from using electrical current to carry the signals in a traditional PCB if it meant you could work with significantly reduced heat? Or is my rudimentary understanding of the physics here betraying me? Friction is going to exist, so it's quite possible that these types of organic fluid logic gates would be useful for other reasons, but not in the ways I've imagined.

Without some(all?) of the electromagnetic changes associated with the continued reduction of die size for traditional electrical circuits, can you build significantly denser (and eventually more powerful) mechanical circuits with fluid-filled cellulose nanotubes?

It's prone to rotting in the presence of fluid, if you make it wet I think that would be a quick and messy ending.

Good point. Would an oil-based transport fluid mitigate that to some degree, or is the organic nature of cellulose going to be a liability to hurdle to making a durable component regardless of the fluid used?

The problems are size and shape change as it saturates and losing structural integrity (soggy...). Capacitors are a nice example, they're always wrapped in something else both to contain the fluid and to keep things in one place.

Any kind of acidity would also be taboo. It would be very hard I think to make something like this but I'm more than willing to admit that none of these are 'dealbreakers'. That's why I asked about the application, after all, if that's the only way to do it and the application is valuable enough then maybe there is a reason to invest a large amount of $ for R&D.