I would imagine the DCC suffers the same limitations as the regular correlation with non-normal data. That is, there isn't an assumption of normality, but non-normal data can cause odd findings; see the Anscombe quartet, for example.
As for kurtosis, taking the log can certainly make it worse. Take this example of the uniform distribution:
x <- runif(100)
logx <- log(x)
where a Normally distributed variable has kurtosis of 3.
on the other hand, in this example
z <- c(rnorm(1000, 10, 1), rnorm(1000, 10, .01))
However, you mention skewed data with kurtosis. Was your data right skew or left skew? Since the former is more common, I'll guess that.
x <- c(rnorm(1000, 10, 1), rnorm(300, 30, 2), runif(10, 500, 600))
Here, taking the log improves kurtosis and skewness.
Taking the log had almost no effect on kurtosis.
As always, try plotting the data to see what is going on in your correlation.