Short answer: Very non-robust. The correlation is a measure of linear dependence, and when one variable can’t be written as a linear function of the other (and still have the given marginal distribution), you can’t have perfect (positive or negative) correlation. In fact, the possible correlations values can be severely restricted.
The problem is that while the population correlation is always between $-1$ and $1$, the exact range attainable heavily depends on the marginal distributions. A quick proof and demonstration:
Attainable range of the correlation
If $(X,Y)$ has the distribution function $H$ and marginal distribution functions $F$ and $G$, there exists some rather nice upper and lower bounds for $H$,
$$
H_-(x,y) \leq H(x,y) \leq H_+(x,y),
$$
called Fréchet bounds. These are
$$
\begin{aligned}
H_-(x,y) &= \max(F(x) + G(y)-1, 0)\\
H_+(x,y) &= \min(F(x), G(y)).
\end{aligned}
$$
(Try to prove it; it’s not very difficult.)
The bounds are themselves distribution functions. Let $U$ have a uniform distribution. The upper bound is the distribution function of $(X,Y)=(F^-(U), G^-(U))$ and the lower bound is the distribution function of $(F^-(-U), G^-(1-U))$.
Now, using this variant on the formula for the covariance,
$$
\mathop{\textrm{Cov}}(X,Y)=\iint H(x,y)-F(x)G(y) \mathop{\mathrm d\!}x \mathop{\mathrm d\!}y,
$$
we see that we obtain the maximum and minimum correlation when $H$ is equal to $H_+$ and $H_-$, respectively, i.e., when $Y$ is a (positively or negatively, respectively) monotone function of $X$.
Examples
Here are a few examples (without proofs):
When $X$ and $Y$ are normally distributed, we obtain the maximum and minimum when $(X,Y)$ has the usual bivariate normal distribution where $Y$ is written as a linear function of $X$. That is, we get the maximum for
$$Y=\mu_Y+\sigma_Y \frac{X-\mu_X}{\sigma_X}.$$
Here the bounds are (of course) $-1$ and $1$, no matter what means and variances $X$ and $Y$ have.
When $X$ and $Y$ have lognormal distributions, the lower bound is never attainable, as that would imply that $Y$ could be written $Y=a-bX$ for some $a$ and positive $b$, and $Y$ can never be negative. There exists (slightly ugly) formulas for the exact bounds, but let me just give a special case. When $X$ and $Y$ have standard lognormal distributions (meaning that when exponentiated, they are standard normal), the attainable range is $[-1/e, 1]\approx [-0.37, 1]$. (In general, the upper bound is also restricted.)
When $X$ has a standard normal distribution and $Y$ has a standard lognormal distribution, the correlation bounds are
$$\pm \frac{1}{\sqrt{e-1}} \approx 0.76.$$
Note that all bounds are for the population correlation. The sample correlation can easily extend outside the bounds, especially for small samples (quick example: sample size of 2).
Estimating the correlation bounds
It’s actually quite easy to estimate the upper and lower bounds on the correlation if you can simulate from the marginal distributions. For the last example above, we can use this R code:
> n = 10^5 # Sample size: 100,000 observations
> x = rnorm(n) # From the standard normal distribution
> y = rlnorm(n) # From the standard lognormal distribution
>
> # Estimated maximum correlation
> cor( sort(x), sort(y) )
0.772
>
> # Estimated minimum correlation
> cor( sort(x), sort(y, decreasing=TRUE) )
−0.769
If we only have actual data and don’t know the marginal distributions, we can still use the above method. It’s not a problem that the variables are dependent as long as the observations pairs are dependent. But it helps to have many observation pairs.
Transforming the data
It is of course possible to transform the data to be (marginally) normally distributed and then calculate the correlation on the transformed data. The problem is one of interpretability. (And why use the normal distribution instead of any other distribution where $Y$ can be a linear function of $X$?) For data that are bivariate normally distributed, the correlation has a nice interpretation (its square is the variance of one variable explained by the other). This is not the case here.
What you’re really doing here is creating a new measure of dependence that does not depend on the marginal distributions; i.e., you are creating a copula-based measure of dependence. There already exists several such measure, Spearman’s ρ and Kendall’s τ being the most well-known. (If you’re really interested in dependence concepts, it’s not a bad idea to look into copulas.)
In conclusion
Some final thoughts and advice: Just looking at the correlation has one big problem: It makes you stop thinking. Looking at scatter plots, on the other hand, often makes you start thinking. My main advice would therefore be to examine scatter plots and try to model dependence explicitly.
That said, if you need a simple correlation-like measure, I would just use Spearman’s ρ (and associated confidence interval and tests). Its range is not restricted. But be very aware of non-monotone dependence. The Wikipedia article on correlation has a couple of nice plots illustrating potential problems.
