"Up" gets into some idioms that are hard to explain. For instance "I won't put up with that."

This calls to mind that when something--a jar, or a notebook, maybe--is put up, that carries the connotation that it is shut away, behind a closed door perhaps, and won't be brought out into the open in the immediate future.
So maybe when someone says they won't put up with that, they
mean that they DO intend to bring it out into the open, and soon.

And that reminds me of 'put up or shut up'.