Does Violating Cosmic Censorship Really Mean Violating Causality? As I understand it, the basic motivation behind ruling out a naked singularity is that we don't know what is happening at a singularity and thus, we won't be able to predict anything in the universe if there is no horizon around such an unknown region. But the reason we don't understand what is happening at the singularity is that we don't have a theory of quantum gravity. But when we have a theory of quantum gravity, this limitation should go away. And thus, causality should be preserved even with naked singularities.
It is a very cultural fact that we don't know how to deal with singularities without horizons at this stage. Thus, it seems quite naive to assume that causality would actually be violated if horizons don't cover the singularity. Though, I believe under some restricted energy conditions the censorship conjecture has been proven and thus, the censorship might be correct due to some other than causality reasons but causality doesn't seem to force the censorship.
 A: The reason naked singularities are a problem is not that they imply causality violation in the sense of closed timelike curves existing (although sometimes they do: see below), it is that they imply that GR is not a useful theory, even in the cases where it ought to be useful, because the future can't be predicted from the past in many cases.  So, in particular, if GR predicts that uncensored singularities arise when starting from physically-reasonable initial conditions, then GR is not useful at predicting what happens in those cases: you need a better theory which makes useful predictions about what happens when GR predicts a singularity.
If cosmic censorship fails, then GR thus fails to be a usefully predictive theory in many cases.  In particular it ceases (or may cease) to be a usefully predictive theory for cosmology.  Well, we would like it to be useful for cosmology of course.
So the question that cosmic censorship seeks to answer is 'is GR, which we know is not a completely correct theory, still usable in the regimes where we would like it to be a good approximation, or does it fail even there?'.
Note that a reasonable (indeed common) definition of 'causality violating' is 'usefully predictive', as Ben Crowell says in a comment: in that sense naked singularities always violate causality.
However it is actually worse than that.  As mentioned in other answers some solutions (Kerr) can have both naked singularities and CTCs while some (Reissner-Nordström) have only naked singularities.
But these are two different pathologies.  So it is not sufficient to have some QG theory which fixes the singularities: that theory would also need to fix the CTCs.
A: There are closed timelike curves in the interior of the Kerr horizon.  The obvious way to see this is if you go through the center of the ring singularity (thus, not intersecting the ring singularity), the Boyer-Lindquist $r$ goes negative, and the Boyer-Lindquist $\phi$ becomes timelike.  Since, by construction, the orbits of $\phi$ are closed, this means that they are closed timelike curves.
A: To my knowledge a naked singularity doesn't imply closed time like curves or other alteration of the ordering of events.
I agree with the OP that a primary example is an overcharged Reisser-Nordstrom.
Still, a naked singularity is a problem, so an actual theory of quantum gravity will need to remove this pathologies.
To be more explicit, a naked singularity means that the space it's not globally hyperbolic, that is there isn't a Cauchy surface, that is given a set of valid and complete initial conditions I cannot predict the future, since singularities act as disturbance points in your equations. See Wald for more info.  
I personally found solutions to supergravity (related to some string theory configurations of branes) with the same asymptotic charges of a naked singularity, but without actual singularities (https://arxiv.org/pdf/1701.05520.pdf, but it's technical, you have been warned!).
