If the true data generating process is ARIMA+GARCH, ignoring the GARCH part and estimating only a regular ARIMA (with constant conditional variance) means estimating a misspecified model. This will yield inconsistent estimates of the ARIMA coefficients and all the consequent troubles. That means the forecasts from an ARIMA models (with a neglected GARCH part) will likely be poor.
In case you had a linear regression with GARCH errors, you would still get consistent (although inefficient) estimates of the linear regression coefficients. Unfortunately, this is not true for ARIMA+GARCH models unless the MA part is empty (the MA order is zero). That is, if you have an AR+GARCH model you may ignore GARCH errors at the expense of loss of efficiency, but still get consistent AR coefficient estimates. That means the forecasts from such a model may be OK, but you could do even better if you actually accounted for the GARCH pattern in the residuals.
If the true data generating process is not exactly ARIMA+GARCH but could be well approximated by ARIMA+GARCH, similar conclusions apply.