Does the concordance correlation coefficient make linearity or monotone assumptions? Does Lin's concordance correlation coefficient assume that the 2 data sets have linear or monotonic tendencies? Or, can I measure the concordance between 2 data sets that have a sinusoidal tendency?
 A: The concordance correlation can be thought of as a measure of agreement. The question is: do two variables $x$ and $y$ (say) have identical values? If so, the concordance correlation will be 1. The question makes no sense unless the variables have the same units of measurement or more generally are recorded in the same way. 
You can calculate a concordance correlation for any variables you like, but the answer will be of no use unless your question is about agreement. You could have a deterministic relation $y = \sin x$, but concordance between $y$ and $x$ will be a meaningless number, if only because concordance correlation does not adjust for different units. 
For an informal introduction to this area, see
Cox, N.J. 2006. Assessing agreement of measurements and predictions in geomorphology. Geomorphology 76: 332-346. http://www.sciencedirect.com/science/article/pii/S0169555X05003740
Here "in geomorphology" indicates the field of the examples, not a restriction of statistical scope. 
A: This statistic measures a kind of correlation between two sets of data.  Its calculation requires no assumptions about what their scatterplot looks like (if that's what you mean by "tendency").
A: If your data are a sample of pairs $(x_i,y_i)$ independently and identically distributed (according to some bivariate distribution); then the sample CCC is an estimate of the population CCC defined as 
$$ \rho_c = \frac{2\sigma_{xy}}{\sigma_x^2+\sigma_y^2+(\mu_x-\mu_y)^2}$$ which: 


*

*equals $0$ when the covariance $\sigma_{xy}$ of the assumed bivariate distribution is $0$;

*equals $1$ when $x_i=y_i$ almost surely.
