There are many papers and book that study this in different settings. One example, would be with Markov Chain Monte Carlo (MCMC) where the samples generated from the posterior distributions are auto-correlated. Time series can be similar. In this case you can measure "effective sample size" which basically measures how much information you have lost by not having proper full random sampling. See for example this [question](http://stats.stackexchange.com/questions/66369/definition-of-autocorrelation-time-for-effective-sample-size?lq=1). 

There won't be a really strong general result for all settings, as there are so many ways to deviate from fully random sampling. 

Note that the opposite is also often true, for example with imbalanced groups stratified sampling can be much more effective then purely random sampling. If you study the effect of age on a dependent variable you might want so sample people so you have a wide range of age.