DIAGNOSIS This is a boot loop, almost always caused by a corrupted software update or a failing internal storage module. The system can't load the operating system completely.
FIX You need to force a factory reset from recovery mode. Here's how: 1) Unplug the TV. 2) Hold down the physical Input and Volume Down buttons on the back of the TV at the same time. 3) While holding them, plug the TV back in. Keep holding for about 15 seconds until you see the Amazon logo, then release. You should see a recovery menu. 4) Use the Volume buttons to navigate to Wipe data/factory reset and press the Power button (on the TV back) to select it. Confirm. This will erase all your apps and logins but can break the loop.
IF THAT DIDN'T WORK If you can't get to the recovery menu, or the reset doesn't stop the loop, the problem is likely hardware—specifically the eMMC storage chip. This is a common point of failure on many Fire TVs after a few years.
HEADS UP A factory reset wipes everything. You'll have to set up the TV like new and re-download all apps. If the loop continues after a successful reset, the hardware is failing and needs professional repair.