Let me take an example based on the estimation of returns to education, which has been a well-studied problem. The usual result is that researchers find the 2SLS estimate to be larger than the OLS estimate by approximately 25%-50%, e.g. Card (1999, 2001).
Three reasons :
An omitted variable that could be negatively correlated with the amount of education. ...
Adding to the excellent answer by @Alecos Papadopoulos, here are two simple numerical examples with $z=x^2+v$, $z^*$ is mean-centred $z$ and $v$ is independent from $u$, in which $E(xu)\neq 0$, $E(xz^*)\neq 0$, but $E(z^*u)=0$.
Example 1 (with $x$ having a symmetric distribution)
$E(xu)=1.5$, $E(xz^*)=-0.5$, $E(z^*u)=0$
Example 2 (showing that the ...