It appears common among meta-analyses to omit studies that do not provide an estimate of sample standard deviation. For example, a previous question reviews some ways to estimate $\sigma$ from different summary statistics.
But is lack of an error estimate (or standard deviation) alone sufficient justification for omitting a study from a meta-analysis? If so, is there a reference for this, and how can this potential source of bias be addressed while implementing a meta-analysis? What are statistical procedures to cope with the bias in estimation of meta-analytic value of effect-sizes and/or variability.