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?
You can test for omitted variable bias without having measurements of the omitted variable if you have an instrumental variable available.
So I'd expand your statement a bit to give:
You cannot test for omitted variable bias except by including potential omitted variables unless one or more instrumental variables are available.
There are assumptions, however, some of them untestable statistically, in saying a variable is an instrumental variable. So if you don't have measurements of a potential omitted variable, you can't avoid omitted variable bias without making some assumptions.
Simple Example:
If the true relationship is described by:
$$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \varepsilon $$
a regression that is omitting an explanatory variable, for example:
$$ y = \beta_0 + \beta_1x_1 + \varepsilon $$
is suffering from omitted variable bias if
Therefore if you ran $ y = \hat\beta_0 + \hat\beta_1x_1 + \hat\varepsilon $ and $x_2$ is available, you can check if leaving out $x_2$ causes omitted variable bias, by checking the two above conditions. However, I don't think that there is any test that will tells you if your regression is suffering from omitted variable bias by just looking at the data that was used in the regression.
There exists no statistical test that detects omitted variable biases.
However, if you suspect that a neglected variable might potentially cause an omitted variable bias and you have an instrument for this variable, then you can test for OVB for this specific variable.
For a general discussion of the omitted variable bias you can check out the following site:
https://economictheoryblog.com/2018/05/04/omitted-variable-bias/
It contains a quite good discussion on how to address the omitted variables bias in general and what precautionary steps one should take before running a regression.
I think that some of the effect of omitting $x_2$ gets absorbed in the estimates of $β_0$ and $β_1$. The rest gets absorbed in the residuals. I think that a good way to see if $x_2$ has an important effect on y that indicates it should be in the model is to fit the model including just $x_1$ and then plot the residuals vs $x_2$. If there is a relationship rather than random variation $x_2$ is important and its omission causes omitted variable bias.
