Is there any literature on the efficiency/asymptotic efficiency of the variance estimators like sample variance? I can find enough analysis about efficiency of mean estimators but nothing on that of variance. Wiki and some other pages delve into variance of sample variance estimators, but don't extend that discussion to relative efficiency.
There is a literature on variance estimators. You might like to start with a literature search for Minimum Variance Unbiased Estimator (MVUE).
If the original observations are iid normal and the mean is unknown, then the sample variance is the MVUE of the true variance. This means that it has the lowest possible variance for an unbiased estimator for any possible values of the unknown mean and variance.
The MVUE result follows from results on complete sufficient statistics and can be found in most theoretical statistics textbooks written for mathematical statistics degrees. For example, if I take Casella & Berger (Statistical Inference, Wadsworth 1990) off my shelf, I find that Sufficiency and Unbiasedness is covered in Section 7.3.3 and the normal variance estimator in particular is covered in Example 7.3.6. Many textbooks cover sufficiency, and the results apply immediately to the normal variance estimator even if this is not explicitly discussed.
Other variance estimators have been proposed if one wants to increase robustness with respect to outliers or to trade off bias for a reduction in mean-square-error. The theory for these variants is more advanced and is exposited mainly in research journals rather than in textbooks.