Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is (weakly) conformal and $f:\mathcal{N}\rightarrow\mathcal{P}$ is harmonic, then $f\circ\phi:\mathcal{M}\rightarrow\mathcal{P}$ is harmonic. (See this book for a reference)

But, I can't find any references that reverse this relationship. Namely, if $f:\mathcal{M}\rightarrow\mathcal{N}$ is harmonic and $\phi:\mathcal{N}\rightarrow\mathcal{P}$ is (weakly) conformal, then is $\phi\circ f:\mathcal{M}\rightarrow\mathcal{P}$ harmonic?

affine structure(i.e., a connection) on $\mathcal{N}$. That's why the two compositions behave so differently. For $\phi:\mathcal{N}\to\mathcal{P}$ to preserve harmonicity in compositions $\phi\circ f$, you need $\phi$ to 'respect'affine structureson $\mathcal{N}$ and $\mathcal{P}$, not necessarily metric structures. – Robert Bryant Jul 25 '12 at 0:10