There are numerous examples of Bayesian analysis of census data in the book Bayesian Data Analysis by Gelman, Carlin, Rubin, and Stern, especially in chapters 5-8.
In fact, whenever the census data collection mechanism is not ignorable, this can often be a crucial part of the analysis. Consider, for example, prior beliefs about demographics that might be under-reported due to unusual household living situations. If you were targeting only a few specific covariates from the census data to estimate something about migrant workers, say, taking this into account would be extremely important.
That example might not be very realistic, but surely there are other examples with census data that highlight a similar point: Bayesian methods allow you to account for hierarchical aspects of data collection and to concentrate your assumptions into well-articulated priors. This would be important when seeking models for underrepresented demographics or non-ignorable designs.