Have any scientific laws been shown to be "illegal," I mean, incorrect? Would this be largely a matter for mathematics, or would scientific experiment be sufficient to debunk a scientific law?

Mathematics may disclose relationships which in turn may lead to hypotheses, but experimental evidence is always required before a hypothesis is recognized as a theory.

In fact science doesn't want to touch subject areas in which a hypothesis cannot lead to a prediction of the result of an experiment; by following this rule, and designing and performing experiments which can prove the hypothesis wrong , the scientist (and peers, if it gets that far) can see if the hypothesis is 'supported'. (It's still not 'proved').

Take the Redi experiment I described in these threads.

Hypothesis: Maggots arise spontaneously in meat.

Prediction: If meat is covered by gauze, it will develop maggots.

Experiment: Two identical samples were exposed in two identical containers and let to stand in the open air for several days. One container was covered by gauze. The other was not.

The experiment was repeated several times.

Every time, no maggots were seen in the covered samples.

Conclusion: The hypothesis is false.

Lead-in to future research: Could the masses of flies which swarmed over the uncovered meat have anything to do with the maggots? Suggest use of newly-invented microscope for the next experiment, in order to see what's happening in minute detail.