[Verizon Galaxy Note 3 Q&A, Help & Troubleshooting] bftb0:
TL;DR Maybe it is a software issue.

There is something subtle that happens when an unsigned boot image is in the boot partition. This is going to sound kind of strange, but it appears that in some conditions where you would expect the phone to be off, when the bootloader detects an unsigned boot image, it will start the kernel up in some kind of software prison where the kernel is actually running, but unable to complete the normal boot. (See this thread, posts #483-#494 where [MENTION=882465]Zzim[/MENTION] posted a very similar set of observations to yours - again with CM13.)

Just now I repeated this experiment - flashed a unsigned image into boot and observed what happens plugging to either a charger or a PC, and then flashing a signed image into boot, and observing what happens in the same circumstance (in each case plugging done after a battery pull & replacement)

Unsigned boot image: Pull battery, replace battery. Plug an active device such as a PC or a dumb charger into the phone. A STATIC battery icon image appears - not an animation - and then a few seconds later you'll see "Set Warranty Bit: kernel" appear momentarily on the screen. If it's a PC you are plugged into, 15-16 seconds later the USB device port changes to a VID/PID device which is exactly the same as the device ID used during a normal boot. But the screen stays dark - the boot never completes! (I don't know if many minutes or hours later the USB device disappears.) That 15-16 second interval is also exactly the amount of time it normally takes for the boot process to be reaching the spot where it is setting up the USB port.

Signed boot image:: Same as above - Pull battery, replace battery, Plug in PC. A battery ANIMATION appears, no Knox message (expected) but also this: with a PC you are plugged into, no USB activity.

It seems a little weird, but its almost as if the bootloader is executing the kernel, but in a way that it's ramdisk has been spoofed or something so that init() never does anything meaningful. Rather suprising but maybe the kernel actually needs to be alive to control the charge control chip? Or produce a battery animation? That sure seems extreme.

Anyway, my experience is that the temasek/CM13 kernel eats through a lot of battery compared to stock kernels. If this phenomenon is still taking place when you plug to a charger and the charge controller isn't working very well, I suppose that could prevent you from gaining any charge on the battery. But I wouldn't expect even that to be chewing through 2A* ~3v ( = 6W) of power in that state, so there must be something else going on. OTOH, if the non-stock kernel messes with the charge controller somehow, then this mechanism potentially *could* interfere with charging.

I tried booting my phone plugged in to the USB charger but with no battery present, thinking, "well, maybe as an experiment he could restore a stock kernel termporarily using a custom recovery". But no joy, phone does nothing.

If both your batteries are so completely discharged that the phone isn't going to do anything, it would seem you are going to need to figure out how to get one of those batteries charged with enough juice so you could boot TWRP and put a stock boot.img in the boot partition. Seems like you need a friend with a Note 3 or a battery store that can juice up your battery enough to flash a stock ROM. Or there's this: boot into TWRP instead of your ROM (& temasek's kernel) and see if it starts gaining charge more rapidly than if temasek/CM13 was booted.

I've only had some older versions of (temasek) CM13 on my phone, so I can't vouch for recent stuff, but I will say that even though it ate up battery, it never caused me to lose charge while the ROM was up and running and the phone was plugged in to a 2A Samsung OEM wall-wart. (I would think that also your phone would get really HOT if it was really getting 2A of current and still could not keep the battery charged)

Which exact version of CM13 were you one when this started happening? I sort of remember someone over there reporting "massive" battery usage for a recent build.

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