I need help in answering this one, it is an exam question.
Strictly speaking, the residuals of a regression are not white noise. Since each residual is a function of the entire data set, the residuals are lightly correlated.
But ... there's correlation and there's correlation.
Residuals can fail to be "white noise" if:
Bottom line: when the residuals fail to be white noise, a different model should be tried.
Short answer regarding time series regression: If they are not white noise (i.e. they are not normal, not have zero mean or serially autocorrelated), then your model is not fully adequate. Therefore, you should revise your model. Usually (but not always), this means that there is a significant autocorrelation (of some order) among the residuals so you should improve your model.