Normality is not a necessary assumption for forecasting. Rather, it is data stationarity. Even though residuals are adjudged as not normally distributed (by, for example the Jarque Bera Test) forecasting can still be done. But if the series is adjudged as non-stationary (by, for example, the Augmented Dickey Fuller Test) forecasting should not be done because such forecasts would be unreliable. 

I don't agree with the previous answer except for the beginning. Yes, "Normality is not a necessary assumption for forecasting". But, no, stationary is not required. Otherwise, ARIMA and SARIMA would be entirely useless. Normality is, however, necessary for the usual confidence intervals, especially for horizons greater than 1 since they are based on the distribution of a linear combination of random variables.

To answer the second part of the question, "Does the violation of the normality of residuals mean that there are omitted variables (such as AR, MA, SAR or SMA terms)?", it is no. In general, normality cannot be achieved by adding variables. Transforming the variable (e.g. taking logs) can sometimes help but not always. 

"As most seasonal time series exhibit increasing trend and/or seasonal variations, both seasonal and nonseasonal differencing are often used to stabilize the time series. If

the time series has unstable seasonal variations, the predifferencing Box–Cox transformation can be performed before further analysis."

https://www.researchgate.net/publication/4871997_Neural_network_forecasting_for_seasonal_and_trend_time_series

Article Neural network forecasting for seasonal and trend time series