Attributes control charts for counts generally assume that the process being monitored is independent and identically distributed in its in-control state. However, violation of this assumption in practice may significantly degrade a chart's performance and usefulness if the autocorrelation structure is not taken into account. To describe the autocorrelation structure of counts in an in-control process, integer-valued autoregressive moving average process models can be employed. This paper investigates the cumulative sum (CUSUM) control chart for monitoring autocorrelated processes of counts modeled by a Poisson integer-valued autoregressive model of order 1, namely Poisson INAR(14). The CUSUM chart is designed to detect assignable causes affecting the process mean, but also changes in the autocorrelation structure are considered. Exact numerical results obtained through a bivariate Markov chain approach are provided for sustained shifts in any or both of these process parameters. Some numerical results from a simulation study of the residuals' monitoring are also presented. It is shown that the considered CUSUM chart of observations has good overall performance in detecting assignable causes in autocorrelated count processes.