In a presentation today the speaker made the above claim. He said that even if the first stage is mis-specified, the coefficient estimates of the second stage will still be valid. As a lowly graduate student I could not ask for an explanation, so now I begged for your help!
Because OLS is unbiased at the mean. Unless it is dramatically incorrect (biased) it really shouldn't matter much what the functional form is.
However, a poor functional form might to cause inaccuracies (slower convergence).
Poor choice of functional form cannot lead to omitted variable bias. Only the omission of a variable.
Using g(x) instead of f(x) is poor functional form. Using g(x) instead of g(x,y) is an omitted variable.