Just downloaded it and installed the app.
First thing I notice: join us with Facebook is a little flaky. The first time I hit the button, it asked me if I wanted to allow the app to open Facebook and I allowed that, then I pressed "continue as Samuel" in the facebook app. Then it just returned me to the same screen it had before. I pressed "join us with Facebook" a second time, and it did some stuff (the screens changed too quickly for me to examine with Voiceover), and it let me in.
Second thing I noticed, after saying "yes" to download Bryan, it put me on the main-screen of the app right away, before the download had finished. When the download was complete, the message didn't get focused with Voiceover, so it took me a minute to figure out why nothing on the main screen was working. I had to explore around the screen to find and press the OK button.
On the saved tab, everything works. However, I can't figure out how to delete the "men walk on moon" article now that I've read it. If there is a delete button somewhere, Voiceover isn't finding it.
Reading articles seems to work just fine. The close button is identified by Voiceover as "CloseModal" rather than just "close", but other than that, everything here is labelled OK. I might strip off the word button from each label though, because at the moment, voiceover is saying things like "ButtonPlay button" "ButtonJumpPrev button", etc, when all I really need is "JumpPrev button". After pressing close on this screen, the play button becomes labelled "ButtonPlaySmall". That's fine, but it could probably stand to be just "play".
On the discover tab, though, things aren't in nearly as good shape. After pressing the personalize button, I can't choose favourite channels. All of the options on this screen voiceover just reads as "button, button, button, button" and so-on. Each of these buttons needs to be labelled. Selecting "top stories" isn't any better. This tab is just full of buttons that haven't been given any labels, either.
On the settings tab, everything works perfectly. No changes at all are needed.
Lastly, it sounds like you're using the Ivona text to speech voices. Are you aware that those voices will be
$1 on December 24 2016? You might want to change TTS providers in the near future. My favourite, by far, are the Neospeech voiceware voices. Though they're also the most expensive. So you might want to consider Acapela or Cepstral instead.
Hope this helps!