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?

Recommendations are for free, so people are always happy to recommend doing more work when they are not the ones who have to do it. That being said, plotting data, including your predictions, is a good practice that will help you notice problems with the data that you have no way to know based on numbers alone (see the famous Anscombe's quartet, for example). Typing plot(resid(model)) or plot(predicted, observed) or plot(observed-predicted, observed)  is really not that much work.
Of course, if you have 100 000 models, one for each gene, or each voxel in a brain, then nobody is checking residuals. (but maybe they should)
Edit, answer to comment: Many of the assumptions are important for inference but not for predictions, so if assumptions are violated but predictions are good, then it's not really a big deal in a sense that it won't make predictions invalid.
However, if residuals are not independent, that means that there is an effect in the data that you can model better, hence improving your predictions. If they are not normal, then you might consider using a more robust model., which would again improve your predictions (according to some metrics). Finally, if they are not homoskedastic, then you can sometimes improve model fit by modeling variance, or you can realize that you are using the wrong model e.g., Gaussian, instead of Poisson/logistic, and you can fix it and again improve your predictions.
I am not saying it will always make a large difference or that you must satisfy your assumptions, but sometimes it helps.
