Why does this paper say that 0-1 loss is insensitive to scaling of weights in a neural network?
Zero-one loss only really makes sense for (hard) classification, so the output of the network needs to already include a discretization.  Presumably the authors have a particular system in mind, and scaling weights doesn't affect the discretized predictions.

Two possibilities (though very closely related) come to mind, though I can't tell whether the authors had either of them in mind.  If the final output of the model is on a linear scale, and the decision is based on the sign of the output, then scaling weights will not change the sign and hence not the decision.  Similarly, if the final output of the model is on a probability scale via a sigmoid activation, then scaling the weights will not change the log-odds's sign, hence will preserve whether the output is above or below 0.5, hence not change the final decision (assuming a 0.5 cutoff).