Does concordance correlation require data to be normally distributed? I would like to know if the results from Device 'A' and Device 'B' are in agreement when measuring the same thing. 
I have 258 pairs of results. The results are interval variables and have the same unit. However, both sets of results are clearly not normally distributed. 
Is concordance correlation coefficient the appropriate test to use?
 A: Concordance correlation is of some use when two methods should agree in their results and the interest is in how closely they do agree. It's then in the first instance a descriptive measure. In practice, what is often even more useful is to look at the fine structure of agreement and disagreement: a plot of difference between variables versus their mean is the place to start. 
It's unclear precisely what hypothesis you have in mind when you imply that concordance correlation is also a test statistic. The hypothesis of zero concordance correlation is usually superfluous, if not absurd, whenever close similarity is expected; conversely, if it cannot be rejected, then you really are in trouble. A hypothesis of concordance correlation being perfect might be of more interest. 
Either way, there is no assumption of normal distributions being made for concordance correlation to be calculated. However, concordance correlation is loosely similar to (Pearson) correlation in that it is likely to be sensitive, directly or indirectly, to skewness, long tails and outliers. It's tacit also that for two variables, $x$ and $y$ say, additive errors $x - y$ of given size should seem of equal importance regardless of the magnitude of $x$ and $y$; that is, a difference between 1 and 2 and a difference between 100 and 101 should seem of equal consequence. If that is not true, then it is likely that you should be calculating concordance correlation for variables on a different scale, most obviously logarithmic scale if errors are better thought of proportionately or multiplicatively. So, as often, a reason for transformation could be given by the flavour of the statistical problem, rather than the shape of the marginal distributions. 
If inferences are to be entertained, I would start with bootstrapping any way, provided that it did not violate any other known feature of the situation.  
