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 $(\Delta u)\circ\psi$ if $|f'|\ne1$ (notice that if $|f'|\equiv1$, then $f$ is an affine isometry). 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.

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 $(\Delta u)\circ\psi$ if $|f'|\ne1$ (notice that if $|f'|\equiv1$, then $f$ is an affine isometry). Hence $\psi$ does not preserve harmonicity. More generally, a function $\phi$ preserves harmonicity ($u$ harmonic implies $u\circ \phi$ is harmonic) if and only if ${\rm D}\phi(x)$ is a similitude, that is the product $\rho(x)R_x$ of some isometry and of a homothety.

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. Edit (after comments below): Liouville's Theorem actually says that a direct $C^4$ conformal map is the composition of an affine similarity and possibly of inversions $x\mapsto a\|x-x_0\|^{-2}(x-x_0)$.

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.

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 $(\Delta u)\circ\psi$ if $|f'|\ne1$ (notice that if $|f'|\equiv1$, then $f$ is an affine isometry). 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.