From my reading (going on memory here) it works as a stand-in electron donor in the electron transport chain in the cell. Cytochrome-C is the 'normal' donor, but this can be downregulated as the mitochondrial population goes down (as it would in an energy crisis). Adding another donor seems to help reverse this. There's some evidence that it can slow the progression of several kinds of dementia.
I know nearly nothing at all about neuropathy, but to the extent that it's from nerve cells in energy crisis, there's a good chance it can help.
For anyone considering this, get some powdered vitamin C, mix some into a glass of water, then add the methylene blue. The ascorbic acid (vit C) will turn the water clear (looks pretty cool as each drop dissipates). Your body will pull them back apart so you'll still pee blue (or green - particularly if you're taking synthetic B vitamins) but you won't be staining your tongue or teeth.
Also, it's CRITICAL that you only take the pharmaceutical version of methyline blue - there are heavy metal poisons in the commercial (for dyeworks) and laboratory (for staining tissues) versions. Only the pharmaceutical version has these removed.
One of the more common off-label uses for MB is 'chemo-brain' (foggy thinking from chemo).