In the context of the "occupy" protests, I've head a couple of times that you should carry a bandana or cloth soaked in vinegar with you, and in case the police uses tear gas/irritants, you should breath through it, or cover your face with it.

Here is one recent source.

Setting aside legal and moral aspects of preparing for such a situation, say you have a good reason... (and also the fact that is stinks...):

Does it work? If so, how?

I'd actually expect the contrary. The vinegar vapors would open up your airways and you would be incapacitated even faster.