Says my Note 10 2014 isn't supported yet. Like @dbrueck stated, it does work on my Note 2 as well.
BUT, I did get it working on my Note 10 Tablet.
I copied both the Amazon Store apk and the video player apk files from my Note 2 over and side loaded them on to my Note 10 tablet. I loaded up the store and went to the videos and it worked. I didn't buy or rent but the movie trailers play which didn't before. Movie trailer playback was smooth on both devices.
Make sure you're using the NON-TABLET version of the Amazon Store app (https://play.google.com/store/apps/details?id=com.amazon.mShop.android)
Also my Note 10 tablet is NOT rooted. Just my Note 2.
The APK files.
Amazon Store (non-tablet version): http://goo.gl/EYRZxX (com.amazon.mShop.android-1.apk version 5.0.0 (500000))
Amazon Instant Video Player: http://goo.gl/qvjJCN (com.amazon.avod.thirdpartyclient-1.apk version 1.0 (1))