Make sure that their Fire TV device and phone are connected to the same Wi-Fi network.
From an Android Device
First, open the Playbook app.
Look for a video you want to watch and hit Play to start watching.
On the top left, click the Cast button.
A window will appear with the names of the device. Tap on "Fire TV Stick".
Your video will appear on your Fire TV screen.
Tap on "Disconnect" to stop casting if you wish to end the casting preview or switch to another video.
From an iOS Device
Launch the Playbook app and play the video that you want to cast.
Tap on the Cast button located in the top left and select "Fire TV Stick" from the list of your available devices.
The video will then appear on your Fire TV screen. All of the playback controls will be available on your Playbook app.
To end casting or switch to another video, tap on "Disconnect" and close the casting window." (as on iOS user cannot close the casting UI without tapping on "Disconnect" first)
NOTE:
Please make sure to enable the Local Network on your device. You will find that option under the Settings section.
If you’re trouble with casting, try casting to web: see video.