When fitting a linear regression model, is it always recommended to plot the residuals? If my linear regression model provides good results on the test set, and my main goal is to predict correctly, is there still a reason to plot the residuals and check the residual assumptions? (for example, to check if they are independent)
Let's assume the regression model has good results for a relevant metric for my goal. When I plot the residuals I see that the residuals are not independent and they do not have a constant variance. When I plot them using QQ-PLOT I see that they are not normally distributed. O.K, many important assumptions are violated. Should I care about it if the model's predictions are accurate?

Another good reason to plot residuals is to check the linearity assumptions. If the residuals are similar whatever the predicted value, then your model seems good.
If the residuals are small for small predicted values, and large for large predicted values, then the assumption of linearity does not seem good. In that case, I would try predicting the log of the Y value instead.
Another thing to look at is whether the residuals are normally distributed - if not, then again there might be ways to make a better model.
