Originally Posted by RayCJ
When you plug the HDMI into the Internet TV, a CPU-based software component has to extract the audio, video and synch signals from the HDMI interface. This is done with software interrupts generated by the HDMI hardware. It then sends the synchronized data to some other CPU-based software component that interacts with the OS to display the output via the video card. If the Internet TV's CPU is messed-up for any one of a number of unrelated reasons, it will prevent those software process from doing their job. In my example case, the Amazon Firestick was configured properly. It was receiving the wireless signal just fine and was presenting the HDMI to the TV's HDMI interface just fine. The TV's own network interface configuration got messed-up somehow and DHCP was releasing the IP address every couple seconds. This caused the Internet TV's CPU to go wild and therefore, it could not process the firestick's signal.
In the case of an older Non Internet-Based TV, the HDMI signal is processed by fixed hardware (most likely a 3rd party ASIC device) and it's less likely to have such problems because the "CPU" in such devices is not involved in processing the HDMI signal.
This may depend on the TV. On my Samsung, before the integrated system has even booted (can't start any of the apps for the first few seconds from cold) you can still cycle through the inputs like a traditional TV and they will all operate. This is an architecture more similar to a traditional TV with a standalone integrated system that just hangs off an unassigned input that gets called when you fire it up via the remote. This is also evidenced by two very different menu systems, one that is very traditional, for cycling inputs, making basic settings changes...etc that is almost identical to what you'll find on a Samsung "traditional" TV, and those that are part of the integrated "Smart" features, which launch apps. Note that this is an older 3D television as well, so that may play into this.
While I get the angle you are approaching this from, the more likely suspect in the OP's case, since he's stated he's used both the AppleTV and Amazon Fire TV on two separate televisions, unless they are identical make/model (hence my query to him on that) is something to do with his selection of streaming service, as that seems to be the common thread. If it works fine with a trial of Netflix, that reinforces that theory.