What is the best strategy for weighting cases with missing values? Using multiple imputation produces multiple datasets. Do I impute values first and then weight cases for each dataset?

That is an interesting question. You might find this article by Andrew Gelman interesting, although it does not exactly address this question.

http://www.stat.columbia.edu/~gelman/arm/missing.pdf

What is your purpose for weighting, and how are you planning to calculate the weights? Raking? Something else?

The dilemma is, I suppose, that some of the variables you would use to calculate weights themselves have missing values. Otherwise, you would just calculate weights first.

I am not sure whether the pooling formulas used in MI would account for variance due to weight differences in the imputed samples. You can, with raking, adjust the control totals so that the nonmissing cases would weight up to the adjusted totals.

Besides weighting the MI datasets separately (after MI), another approach would be to apply single imputation, then construct a single set of weights, and then discard the singly imputed values and do MI with the weights that resulted.

I can't say which approach is best, but if you are using weights for model building, as opposed to estimating population totals, it would be wise to try the different approaches and see whether they affect the conclusions. If the conclusions are robust to the weighting and imputation strategy, you are in good shape. If not, you may need to dig deeper into the relationship between missingness and weighting.