I heard that a reason for the low energy consumption of a kindle is that it uses some sort of "electronic" ink. Electricity is then only used to change the ink patterns. This made me wonder if you could use a kindle as boarding pass. I have tried boarding with an iPod touch by storing the pdf on it, but it was not accepted. I had to fetch a paper print of my boarding pass.
I think it depends on the airline you're travelling with, and from what airport.
Some airlines have launched mobile boarding passes (exact name may vary between airlines). With this, you're able to call up your boarding pass on your phone, and the barcode on it can be scanned in at the various points where a boarding pass is needed. As the airport infrastructure needs to updated to be able to read mobile boarding passes, it isn't available at all destinations of the airline.
Typically, this is done via an application on your phone. The app lets you pull up your boarding card, allows it to be updated if requested, and ensures that the information is presented in the required size and format. In some cases though, I think it's done with a PDF.
For the Kindle case, if your airline supports mobile boarding passes, then I'd expect you'll be fine to use it. Just ensure that the 2D barcode, the flight number and your name all show on the screen at the same time, so everything that's needed by a human and by the barcode scanner are all available at once.
If your airline doesn't support mobile boarding passes, then you're likely out of luck. Even if they use 2D barcodes on their printed boarding passes, the barcode reader could well have issues with reading from the e-ink screen, so you're unlikely to be able to use it. The only good news is that more and more airlines are rolling out mobile boarding passes (as they're more convenient for users, and more convenient for pushing out changes than re-printing boarding cards), and more airports are supporting them.