Let $f : \Delta \rightarrow \mathbb{C}$ be a holomorphic function on the unit disk. Let $u = |f|$. Assume that $u(0)=f(0)=0$.
Question : since $u$ is harmonic, the mean value property should imply that $u(0)$ is the average of $u=|f|$ on say a disk of radius $1/2$. But since $u(0)=0$, this would imply that $u=0$ on a neighborhood of $0$ and therefore $f=0$. Clearly I am making a false reasoning here. Sanity check : what is the problem ? I would say it is the fact that $u$ is harmonic, but I'm not sure.
As Daniel Fischer said, $|f|$ is not harmonic. One way to see this is by calculation of Laplacian of $|z^2| = x^2+y^2$. The Laplacian is $4$, not $0$.
It is true that $\log|f|$ is harmonic on the set $\{f\ne 0\}$.