What would happen if the kernel was somehow modified to avoid sending the shutdown signal? I'm assuming a signal is sent when the battery reaches some critical level.

If that signal was programmed to be ignored somehow, what would me most likely to happen when the battery depleted its charge?

Would it be the same as plainly removing it, with the screen going off?

My guess is that the memory spaces wouldn't be properly managed, so what would happen when the device was turn on again?

Sorry if this seems too vague.
If the battery depleted without a phone shutdown call, it would essentially be the same as a battery pull, where the power is suddenly cut off from the device.

One of the major risks is preventing this automatic shutdown is data loss/corruption, but this should only occur if you were in the middle of a data process (eg. saving offline document, camera/screenshot).