ARMA-GARCH, invertibility, stationarity and insignificance 
I am trying to forecast volatility out-of-sample using ARCH, GARCH, GJR and EGARCH. I used AIC to identify the ARMA and ARCH order and decided to stick with (1,1) for GARCH-type models. However, I have situation where my mean equation is either insignificant and/or non-invertible and non-stationary. The insignificance does not worry me too much, but I think the non-invertibility/non-stationarity will affect my forecast results out-of-sample. Should I remove the mean equation in the models where this happens? If no then how would this affect my forecast results out-of-sample?
 A: I agree that nonstationarity and noninvertibility are the two more serious concerns when it comes to forecasting. If you have a nonstationary process where nonstationarity is due to the level of the series (the unconditional expectation is undefined or changing over time), you will have trouble forecasting the level of the series. 
Is that important when you are actually interested in forecasting volatility? Yes, it is. You either define volatility as the magnitude of deviations from the conditional mean (the level) or from zero. In both cases nonstationarity in mean will cause you trouble. 

Should I remove the mean equation in the models where this happens?

Removing the mean equation will not solve the problem. In fact, you cannot remove the mean equation. A GARCH model specifies the conditional distribution of the time series, and that by definition specifies the conditional mean (unless it does not exist). In a GARCH(r,s) model for a time series $y_t$,
\begin{aligned}
y_t &= \mu_t + u_t, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2. \\
\varepsilon_t &\sim i.i.d(0,1), \\
\end{aligned}
you have the element $\mu_t$. It can be zero, it can be something more complicated, but it is there, it cannot just be removed. And it specifies the conditional mean of the distribution of $y_t$, the latter being $y_t \sim D(\mu_t,\sigma_t^2)$. (It has to do with the properties of a random variable. Ignoring some property of a distribution, such as the mean, does not make it disappear.)
What I suggest is trying to see why the series appears to be nonstationary and address that (by adjusting for a deterministic or a stochastic trend, a structural change or the like). Then you will have a solid ground for your volatility forecasting.

If no then how would this affect my forecast results out-of-sample?

If the conditional mean model is well specified and estimated well (e.g. the true process is explosive like $y_t = 1.1 y_{t-1} + u_t$ and it is estimated as such), your forecasts of the conditional mean should be fine. On the other hand, if there is, say, a neglected structural change, then the model will be misspecified and the forecasts will likely be poor.
