The Apple Lightning to SD Card Camera Reader does not work with the iPhone.
Even if the clerk at the Apple Store tells you it works great with the iPhone, and you can use it to pull photos off an SD Card and onto your iPhone for further manipulation and uploading, it still doesn’t work. Plugging it in gives you a “This accessory is not supported by iPhone.” message.
I point this out here because it seems to be a common question without an answer on the Apple forums.
Much to their credit, the Apple Store gave me a full refund (no restocking fee) on the device, even though I was outside of the return period by one day. (I bought it a little over two weeks ago, but had not taken it out of the package until yesterday because I hadn’t shot any photos until Sunday.)