If you're accessing Facebook through the Kindle Fire HD's web browser, some games won't work due to them using Adobe Flash. Unfortunately, Adobe discontinued Flash support for mobile devices. The Kindle Fire 1st Generation still has Flash, but that is because it was released when Adobe's support was still intact.
If you're accessing the game through its own app and are unable to connect your Facebook account through it, the following troubleshooting method may help:
1. Swipe down from the top
2. Tap the "More" button
3. Select "Applications"
4. Choose "Installed Applications"
5. Pick your game
6. Tap "Clear Data"
You'll then want to restart your Kindle Fire device. Make sure your Kindle is unplugged and then push and hold the power button for 20 seconds (ensure you ignore the Shut Down notice). Release the power button and wait a few moments before pushing it again.
Since Facebook connectivity goes through Facebook's servers, you may also want to reach out to them through their help page.