Are there any disadvantages to state*year fixed effects? Suppose that my unit of treatment is the county, and that counties in multiple states receive the treatment and suppose that I have panel data. Should I always use state*year fixed effects to control for state-specific shocks, or would there be disadvantages to this strategy?
For example, I know that there are basically no downsides to using year fixed effects and that I should always use them in panel data. 
However, county fixed effects have disadvantages, for example:

Coefﬁcients of time-varying regressors are estimable, but these
  estimates may be very imprecise if most of the variation in a
  regressor is cross sectional rather than over time.

From Cameron & Trivedi, Microeconometrics Methods and Applications, p715. 
 A: I think it's good practice to try to avoid using terms like "always" or "never" when thinking about problems like this, because the answer is really dependent on your problem. One case where you should add state*year FEs is if you know for sure (either from theory and/or your model, or experience) that states widely and fundamentally differ in responses, and you want to accordingly 'control' for that when studying counties (and counties were not randomized pairwise by state, or some other design). When that is not the case, the answer truly does depend: you should add state-specific fixed effects if you think it's important and your dataset is large enough to handle the inclusion of these fixed effects. The last part is important. What makes you sure that you should add year FEs but not yearXstate FEs? I'd guess changes over time are probably as important as changes across states (again, unless theory/experience tells you otherwise), so the true reason probably has to do with the fact that yearXstate FEs adds (max year - min year)*50 dummy variables. And if year FEs are important, what about seasons? and months? and days? One drawback for all of these is that with small datasets, you run into problems with too many covariates.
In the end, your model is an approximation of reality, and so you want to include what is important, and not add everything (in some sense, though certainly not the only, nor most important reason, this is also one argument for using linear regression). 
Specifically for your problem, without knowing the specifics, I think the standard from what I've seen from similar work is to either add state FEs + year FEs (importantly, not the interaction), just year FEs, or yearXstate FEs. Definitely don't try each and keep what works best and drop the one that is worse, but do think about your problem, the treatment, your outcome, etc, and see if you can convince yourself what is the right approach. If outcome is access to school funding, then since states exercise a fair degree of control over that, maybe you should add it. If outcome is wind levels, then states really don't matter, and you should maybe control for elevation, or something else. And so on.
