We know a linear probability model (LPM) will produce heteroskedastic errors by definition because of how the variance of a bernoulli r.v. is defined. My question is whether the same is true for logit/probit. I know that logit/probit can theoretically be amended so that marginal effects are not biased by heteroskedasticity, but does the argument for why heteroskedasticity is certain in an LPM carry through to the nonlinear logit/probit world?
Logit and probit are special cases of the generalized linear model. (For that matter, so is linear or 'OLS' regression, and thus so is the linear probability model.) The generalized linear model was developed to, um, generalize standard regression methods. The problem with OLS regression, when used for a binary dependent variable, is that it assumes the response is conditionally normal. With a GLM, you can specify the response distribution as binomial. From there, you can specify a link function, even using the identity function if you really want. Logit and probit are simply different link functions (it may help you to read my answer here: Difference between logit and probit models). A GLM with a binomial response, and either of those two link functions will properly account for the fact that the conditional variance changes as a function of the conditional mean. So heteroscedasticity, in that sense, is not necessarily a problem. (Whether you want to call the differing conditional variance 'heteroscedasticity' or not is somewhat a semantic issue.)
That does not necessarily mean that the model is modeling the conditional variance correctly, though. One issue could be that the functional form is wrong, or that you should have used the other link function, etc. Those imply that the conditional mean is incorrect as well. But even beyond that, you can have data that vary too much. If you only have one observation at each value in X, you wouldn't be able to see this, but if you have multiple data, you can sometimes see that the means are bouncing around the predicted means more than they should. This can happen because there are other relevant variables that you haven't included (you may not even know what they are). You could call this 'heteroscedasticity' as well (i.e., the term is somewhat ambiguous in this context), but it would be more common to call it overdispersion.