In general, complete case analysis is biased when data are not MCAR. However, when the analysis consists of fitting a regression model, complete case analysis is unbiased under the weaker condition that missingness is independent of the outcome variable, conditional on the covariates. Depending on which variable(s) contain missing values, this condition sometimes corresponds to MAR mechanisms, and sometimes to MNAR mechanisms.
For example, suppose the outcome Y is the variable with missing values. Then missingness being independent of outcome conditional on covariates corresponds to the MAR assumption, which says the probability of missingness is independent of the partially observed variable conditional on the fully observed variables. Alternatively, suppose that missingness in a covariate X depends on the value of that covariate, so that data are MNAR. Provided missingness in X is independent of Y, conditional on X and other covariates in the analysis model, complete case analysis is unbiased.
For more on this, see:
1) the paper cited in a previous answer by White and Carlin: Bias and efficiency of multiple imputation compared with complete-case analysis for missing covariate values. Ian R.White and John B. Carlin. Statistics in Medicine, Volume 29, Issue 28, 2010
2) a paper by myself and colleagues published in Biostatistics: http://doi.org/10.1093/biostatistics/kxu023
3) a blog post I previously wrote about this here: http://thestatsgeek.com/2013/07/06/when-is-complete-case-analysis-unbiased/