You need to connect your Amazon Fire TV through adb:
After connecting you need to run adb install command:
adb install /Users/...../YourAppPath/android/app/release/app-release.apk
Provide the full path of your apk to adb install command.
adb install /Users/foouser/Projects/MyAndroidApp/android/app/release/app-release.apk
The adb push process will start and the installation process of your apk to your Amazon Fire TV will start:
Performing Push Install [ 20%] /data/local/tmp/app-release.apk ..... ... Success
After the process is done the apk will be installed to your Fire TV.