Standard control charts require that observations from the process are independent of one another. Failure to meet this requirement increases the chance that the control chart will indicate a process shift when the process has NOT shifted (a false alarm). Therefore, the Autocorrelation Function is a good tool to use to check the independence assumption. If control limits on an X-bar chart are particularly tight, with many out of control points, autocorrelation should be suspected.
If the autocorrelation is only significant at low lags (adjacent data points), you can increase the time between acquiring data points to lessen its effect. In other cases, there might be auto correlation due to sampling from multiple streams in a process. For example, when monitoring order processing times, if each data point is the time taken by each one of five employees operating at a different average level, then an autocorrelation would appear at lag 5.