What if error of linear regression is uncorrelated between different observations, but dependent? One of the assumptions of classic linear regression is that error is uncorrelated between different observations. But it is obvious that uncorrelated is not independent. I have already learn the situation of correlated errors. But I am thinking are there any situations that errors are uncorrelated $Cov(\epsilon_i,\epsilon_j)=0$, but also dependent? What should we do for that situation? And how can we detect that?

With uncorrelated errors, the linear regression coefficient estimates and standard errors should still be correct; tests and confidence intervals will not be correct in general ― e.g. the significance level and coverage respectively may differ from what you want ― but in large samples this might not matter all that much.
"Dependence" ― even under the restriction of 0 correlation ― can cover a very wide variety of possibilities. Consequently, if you don't have a specific form of dependence in mind to check for, it can be hard to detect (very large samples may be required to get good power against very broad alternatives).
