UIDocumentInteractionController for Open In menu - Doesn't Work
I found the answer, and it's memory management. I create the UIDocumentInteractionController and then present it, but I don't have it as an instance variable. ARC deallocates it before it has the opportunity to do anything. This includes sending the document to the external app.
This bug didn't appear on the iPhone, but on the iPad it gives an error because the popover architecture works a bit differently and it ends up trying to draw it when it's deallocated. That's what alerted me to the bug.