The "official" PoE/PoE+ hats will not work with the Pi 400 for a couple of reasons: The main reason being that there is no access to any of PoE pins on the Pi 400 PCB, as there are on the Pi 4 board (just behind the Ethernet port) - indeed, ignoring the fact that the case of the Pi 400 prevents any access to the PCB of the Pi 400, the PCB of the Pi 400 doesn't even have these pins.
However, the Pi-Supply PoE Switch Hat (which supports 802.3af) does not rely on these pins as it has its own input Ethernet port, from which the PoE is extracted, and as such is also compatible with the Pi 2.
Being the optimist, I tried one of these PoE hats with the Pi 400 but it didn't work - it doesn't power the board - not even if you hit the power button on the Pi 400, nor click the switch on the PoE PCB.
Fortunately, it doesn't blow either the Pi 400 nor the PoE hat - I have checked that both the Pi 400 and the PoE Switch hat continue to function afterwards (the Hat on a Pi 4, and the Pi 400 powered via the usual USB-C connection).
So, my question is, what is different about the Pi 400 (or rather its PCB) that means it can not be powered by directly applying power to the power pins on the GPIO?
Note I have not tried just applying power to the power pins of the Pi 400 directly using a bench power supply (yet)... I should probably do that. However, before potentially blowing the Pi 400 by doing so, I thought it judicious to check here first to see if anyone knows more about the power circuitry of the Pi 400.