If a time series exhibits autoregressive conditional heteroskedasticity, including a GARCH specification for the conditional variance makes natural sense. Failure to do so may result in inefficient and sometimes (although not always)* inconsistent parameter estimation of the conditional mean model, and also inferior fitted values and forecasts of the conditional variance.
Conversely, in absence of autoregressive conditional heteroskedasticity there is no point in modelling the conditional variance as a GARCH process. It will not, for example, properly account for autocorrelation in the time series if there is any. On the other hand, if both the conditional mean and variance models are misspecified to begin with (e.g. a model is just a constant plus noise, while the true data generating process is something like ARMA-GARCH), it is hard to tell what effects including a GARCH specification (but still leaving the conditional mean equation misspecified) will have.
* For example, if the conditional mean is modelled as an autoregressive (AR) process and the model is estimated using conditional least squares, neglecting conditional heteroskedasticity does not make the parameter estimates inconsistent, because the least squares estimator does not require homoskedasticity for consistency.