Is the sample variance a useful measure for non-normal data? Does it ever make sense to compare the variance for two sets of data, neither of which are even approximately normal (e.g. bimodal)?
 A: Does it ever make sense? I don't know. It depends on what you are trying to do. But you say you aren't trying to do anything, so there's no good answer.
I can't think of a case where the variance of a bimodal distribution makes much sense. But you say (e.g.) bimodal. If the data are non-normal, but not all that non-normal, variance can make sense. My general rule is that, if the mean makes sense, the variance makes sense. If the median makes sense then either the mean absolute deviation or interquartile range makes sense.  For bimodal distributions, neither the mean nor the median makes all that much sense. A density plot is a good method here, or a table of percentiles. 
A: We frequently use scaling relationships between central moments to ascertain the most likely underlying generative process of biological data. I studied the dynamics of the HIV promoter and it produces very non-Gaussian distributions that are wide and highly skewed. The Var~Mean relationship was the first step in determining candidate stochastic models. Moment analysis is never an end in itself, without interpretation they're of limited use. 
