Think about what CV is: Ratio of standard deviation to mean. But if the variable can have positive and negative values, the mean could be very close to 0; thus, CV no longer does what it is supposed to do: That is, give a sense of how big the sd is, compared to the mean.
EDIT: In a comment, I said that if you could sensibly add a constant to the variable, CV wasn't good. Here is an example:
set.seed(239920)
x <- rnorm(100, 10, 2)
min(x)#To check that none are negative
(CVX <- sd(x)/mean(x))
x2 <- x + 10
(CVX2 <- sd(x2)/mean(x2))
x2 is simply x + 10. I think it's intuitively clear that they are equally variable; but CV is different.
A real life example of this would be if x was temperature in degrees C and x2 was temperature in degrees K (although there one could argue that K is the proper scale, since it has a defined 0).
In examples like yours when data differ just additively, i.e. we add some constant $k$ to everything, then as you point out the standard deviation is unchanged, the mean is changed by exactly that constant, and so the coefficient of variation changes from $\sigma / \mu$ to $\sigma / (\mu + k)$, which is neither interesting nor useful.
It's multiplicative change that's interesting and where the coefficient of variation has some use. For multiplying everything by some constant $k$ implies that the coefficient of variation becomes $k \sigma/k \mu$, i.e. remains the same as before. Changing of units of measurement is a case in point, as in the answers of @Aksalal and @Macond.
As the coefficient of variation is unit-free, so also it is dimension-free, as whatever units or dimensions are possessed by the underlying variable are washed out by the division. That makes the coefficient of variation a measure of relative variability, so the relative variability of lengths may be compared with that of weights, and so forth. One field where the coefficient of variation has found some descriptive use is the morphometrics of organism size in biology.
In principle and practice the coefficient of variation is only defined fully and at all useful for variables that are entirely positive. Hence in detail your first sample with a value of $0$ is not an appropriate example. Another way of seeing this is to note that were the mean ever zero the coefficient would be indeterminate and were the mean ever negative the coefficient would be negative, assuming in the latter case that the standard deviation is positive. Either case would make the measure useless as a measure of relative variability, or indeed for any other purpose.
An equivalent statement is that the coefficient of variation is interesting and useful only if logarithms are defined in the usual way for all values, and indeed using coefficients of variation is equivalent to looking at variability of logarithms.
Although it should seem incredible to readers here, I have seen climatological and geographical publications in which the coefficients of variation of Celsius temperatures have puzzled naive scientists who note that coefficients can explode as mean temperatures get close to $0^\circ$C and become negative for mean temperatures below freezing. Even more bizarrely, I have seen suggestions that the problem is solved by using Fahrenheit instead. Conversely, the coefficient of variation is often mentioned correctly as a summary measure defined if and only if measurement scales qualify as ratio scale. As it happens, the coefficient of variation is not especially useful even for temperatures measured in kelvin, but for physical reasons rather than mathematical or statistical.
As in the case of the bizarre examples from climatology, which I leave unreferenced as the authors deserve neither the credit nor the shame, the coefficient of variation has been over-used in some fields. There is occasionally a tendency to regard it as a kind of magic summary measure that encapsulates both mean and standard deviation. This is naturally primitive thinking, as even when the ratio makes sense, the mean and standard deviation cannot be recovered from it.
In statistics the coefficient of variation is a fairly natural parameter if variation follows either the gamma or the lognormal, as may be seen by looking at the form of the coefficient of variation for those distributions.
Although the coefficient of variation can be of some use, in cases where it applies the more useful step is to work on logarithmic scale, either by logarithmic transformation or by using a logarithmic link function in a generalized linear model.
EDIT: If all values are negative, then we can regard the sign as just a convention that can be ignored. Equivalently in that case, $\sigma / |\mu|$ is effectively an identical twin of coefficient of variation.
EDIT 25 May 2020: Good detailed discussion in Simpson, G.G., Roe, A. and Lewontin, R.C. 1960. Quantitative Zoology. New York: Harcourt, Brace, pp.89-94. This text is inevitably dated in several respects, but includes many lucid explanations and pugnacious comments and criticisms.
See also Lewontin, R.C. 1966. On the measurement of relative variability.
Systematic Biology 15: 141–142. https://doi.org/10.2307/sysbio/15.2.141
Best Answer
It seems to me that CV is inappropriate here. I think you may be better off separating the change in location from the change in dispersion. In addition, the distributions you mention in your comment to the question are, for most parameter values, skewed (positively, except for the beta distribution). That makes me question whether the standard deviation is the best choice for a measure of dispersion; perhaps the interquartile range (IQR) might be better, or possibly the median absolute deviation? Similarly, rather than the mean, I might consider the median or the mode as the measure of location. The choice might in practice be determined by ease of computation as well as the field of application, the details of the model...
Say you choose to use the IQR and the mode. You could summarise the change in dispersion using the ratio of posterior to prior IQRs, probably plotted on a log scale as that's usually appropriate for ratios. You could summarise the change in location using the ratio of the difference between posterior and prior modes to the prior IQR, or to the posterior IQR, or perhaps to the geometric mean of the posterior and prior IQRs.
These are just some quick ideas that came to mind. I can't claim any strong underpinnings for them, or even any great personal attachment to them.