I truly dislike the off function of the 4K Firestick. It only turns off the TV but the Firestick is still active and if you have more than two Firesticks you will get a message that only two devices may stream simultaneously (on Recast). My solution which would work for your issue is to use a smart power strip (has a master plug that when turned off will turn off up to 4 other devices plugged into the slave sockets.) When I turn off my TV via its sleep function, all the other devices connected to the TV will power down.
Oddly enough, the fire stick is the only device I have connected to my TV that won't let it go into sleep mode on its own. My Sony Blu Ray player, lets my TV do this, my Xbox 360, my laptop connected, etc. This isn't hard, the fire stick should stop sending a signal when it's in sleep mode. Then the TV itself could go into sleep mode...
The biggest problem with the way it works is that it keeps the backlight on at all times , that's how you burn out your screen and then have to get another backlight or failing that a while other TV, I've had this TV for years and never had any issue with it going into sleep mode ever until now. This is specifically a fire stick issue and really needs to be looked at by the devs... Not be told to just set a sleep timer or turn it off. It does this automatically when it's not receiving a signal.
You could purchase a smart power strip that when the master device is turned off all the subordinate devices are turned off. I use it on 6 fire sticks and it only fails to work (does not power off the firestick) when it is plugged into my AV receiver.
i have a fire stick not the fire TV but it behaves the same way. Like the orrigional post, and some other posts state, when my Fire Stick goes to sleep mode (after the screen saver) it sends blank screen to the TV instead of sending no signal through the HDMI. I have read some of the suggestions posted but they really are not solutions and don't seem to understand why people want the sleep mode to turn off the HDMI. in this house this is what happens, start something on the firestick, settle into the couch (or bed), watch some stuff, doze to sleep. Wake up many hours later, Stick has gone to sleep but since in sleep mode it sends blank screen signal, it prevents the TV from doing its sleep mode stuff. Its been on all the time. Its also very possible to get up and walk away (semi groggy of course, just woke up!) leaving the TV on because it seems dark. This is not desirable as TV will age faster, consume more power, etc. This is exactly the reason why a few years ago regulations were passed to have all new TVs power down after getting no signal through HDMI after a period of time. Sending a blank screen instead of turning off HDMI, fire sticks and fire TVs bypass this. I am interested if anyone has an answer to the posted question. Messing with a TV sleep timer is not really a solution, I often dont know how long before i end up drifting to sleep.