Your "examples" usually don't work. Up to a complex conjugation (= symmetry), a $2$D-mapping $\phi$ that preserves harmonicity is a holomorphic function, *via* $z=x_1+ix_2$ and $f(z)=\phi_1+i\phi_2$. Now, let us form $\psi(x_1,x_2,x_3)=(\phi(x_1,x_2),x_3)$. What happens is that
$$(\partial_1^2+\partial_2^2)(u\circ\phi)=|f'|^2(\partial_1^2u+\partial_2^2u)\circ\phi.$$
From this, you see that $\Delta(u\circ\psi)$ is not proportional to $(\Delta u)\circ\psi$. Hence $\psi$ does not preserve harmonicity.

On the contrary, a theorem due to Liouville says that a mapping in ${\mathbb R}^3$ that preserves angles must be an affine similitude. This applies in particular to mappings that preserve harmonicity.