I am aware of the Ramsey Reset test which may detect nonlinear dependencies. However, if you just throw out one of the regression coefficients (merely linear dependencies), you may get a bias, depending on the correlations. This is obviously not detected by the Reset test.
I did not find a test for this case, but this statement: "You cannot test for OVB except by including potential omitted variables". It is probably a reasonable statement, isn't it?