Why do many, if not most contemporary philosophers (especially naturalist analytical ones à la Quine) believe in the existence of a set of unchanging natural laws despite the fact that this assertion has not, and probably cannot, be substantiated? By 'natural laws,' I mean laws like those associated with physics, etc. rather than laws dictating which sorts of inferential deductions are valid/invalid. Would this belief fare better when faced with a Russell's teacup-style argument than theism does?

I'm not sure what, exactly, leads philosophers to believe in "unchanging laws," and I'm also not sure in what sense such laws cannot be substantiated. I'll focus on the latter. If nothing else, can't they be confirmed to some degree? You should expect that a world without any unchanging laws would look one way (perhaps random, unpredictable, inexplicable stuff happening all the time), while a world with unchanging laws would look another way (exhibiting patterns, being susceptible to some sorts of explanations and predictions, etc.). We seem to be living in a world more like that second, and not so much like the first. So, this seems to confirm, or make more likely, the hypothesis that there is a set of fixed laws governing it all. This argument I just rehearsed is, to be sure, debatable! And one immediately recalls Hume's argument about "induction" or the uniformity of nature. And we do keep finding surprising and (momentarily) inexplicable things (though they do tend to be explained eventually, don't they?) But, at least it suggests that there may be something more than mere dogma to people's belief in unchanging laws. That is, at least one can imagine an argument (even if that argument ultimately fails). Perhaps that explains why philosophers believe in unchanging laws, or even justifies the belief.
As for the teacup example, this is a very interesting connection to draw. I am guessing this is what you have in mind: Belief in God is like belief in unchanging laws in that it is not grounded in any argument, and nor, perhaps, can it be disproved. But the fact that it cannot be disproved does not show that we should seriously entertain it. We cannot disprove that there is a teacup orbiting the earth, but we do not seriously entertain that, in fact we regard it is extremely unlikely. So, we may also regard the belief in God as extremely unlikely and--this is what I think you are suggesting--the belief that there are unchanging laws as unlikely.
But I think unchanging laws fare better than God when it comes to this argument, and I also think the argument itself is flawed. So, double whammy again the teacup argument against unchanging laws!
In the first place, what I suggested above is that there does seem to be at least some reason to think that there are unchanging laws (namely, some of the things we might expect if there were no unchanging laws do not happen). That makes it at least a little more likely that there are unchanging laws. There is no such argument for orbiting tea cups or undetectable Gods. How would we expect things to look if an undetectable God existed? Just like this, he's undetectable. How would we expect things to look if such a God didn't exist? Just like this, because again, he's undetectable. But this same reasoning does not apply to unchanging laws: we have different expectations of what to observe depending on whether there are such laws.
Secondly, I think the teacup argument, even against an undetectable God, is flawed, because the analogy is a bad one. We have lots of reasons for thinking that an orbiting teacup is exceedingly unlikely. We know what teacups are, where they come from, what it would take for it to overcome earth's gravity, what motivations a person might have for spending that much money to launch a teacup, what sorts of things we have found out there so far in orbit, and so on. We have lots of background information that makes an orbiting teacup unlikely. This is why we think it is unlikely, it's not that it is unlikely merely because we lack evidence for it either way. When it comes to an undetectable God, the idea is supposed to be that, not only do we lack direct evidence against it (or a disproof of it), but we also lack that sort of background, indirect evidence that makes it unlikely. So, the analogy falls apart. I'm not suggesting that an undetectable God is likely, or that we should believe it. But I am suggesting that the teacup argument against (belief in) an undetectable God fails. Thus, even if unchanging laws didn't fare better than God against the teacup argument (which, as my first point suggests, they do), that wouldn't mean much for unchanging laws.
If you haven't encountered it yet (which I sort of doubt!), you would probably be interested in Hume's discussion of the Uniformity of Nature (see his Enquiry Concerning Human Understanding) and perhaps Salmon's contemporary discussion of it). Feel free to email me to get those references. Thanks for a great question!

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