Suppose the true model is
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon$$
where $x_1$ and $x_2$ are correlated and $\epsilon$ is white noise. I omit variable $x_2$ and apply OLS to estimate
$$y = \beta_0 + \beta_1 x_1 + \epsilon$$
Now my predictor $x_1$ is correlated with the noise ($\epsilon + \beta_2x_2$), thus the Gauss-Markov assumptions are not satisfied so I can't say the OLS estimator is the Best Linear Unbiased Estimator.
But what about predicting $y$ given only $x_1$? Is OLS still optimal in some sense?