The product page has "A Fire TV streaming media player or Echo Show—must be on the same network and registered to the same Amazon account as Fire TV Recast" and "You cannot use Fire TV Recast to record live TV programs from any other source, such as cable or satellite boxes or streaming apps. You can only register 1 Fire TV Recast per Amazon account.". This quote from further down the page "You can watch live or recorded programs, and you can record shows airing live, when you are on-the-go in the Fire TV mobile app using cellular or Wi-Fi connection.: might seem to indicate you could do what you want; however, I think it is just that from a remote location you could tell your Recast to record (I have no idea how to do it or if it works.)
I watch live TV and recorded shows from my Recast over LTE and other wifi networks all the time. Check that your profile on the Fire HD tablet matches the one registered to your Recast. Open the Amazon shopping app, open the top left menu, select "your account", "manage content and devices", select the "devices" tab, make sure your Recast shows up in the devices drop down. If not, your Recast might be registered to another account in your household.
I also watch my Recast over LTE and WiFi all the time, both on my Note 10 and my Chromebook. So unless there is something specific on the Kindle Fire HD and the version of the app it uses, Chrisrva is on the right track.
I would also add that you need decent upload speeds at home to be able to stream from the Recast. Depending on your ISP and your connections speeds, the upload side of things may not be fast enough to support the stream. You can try to adjust the stream quality on the app if it seems to buffer or just never starts the stream which may help as well.
But as Chrisrva mentions, make sure your logged into the correct Amazon account that your Recast is registered to and that is shows up as available in the app.
All-New Fire HD 10 Tablet, 3 Fire TV sticks ,an older 7" Fire Tablet..and the NEW fire TV Recast.
The only device that does not 'pair up' to work with the Recast is the new Fire Tablet 10. I have downloaded the Fire TV app to the new Tablet and when I go to the app and click 'On Now',
it say 'Please connect to fire TV Recast. I then goto 'Settings' and then to 'Fire TV Recast' and it tells me 'Not Paired'. I have un-pluged and re-stared the Recast and followed the how to, but still nothing. I even wiped the new tablet clean and started from the ground up...so..any other ideas?