This question is quite subtle, I don't believe the answer is known in  general situation. But if you consider the case of $1$-forms on surfaces, one can completely characterise those that are harmonic. They form a "space" of finite dimension modulo self-diffeos of the surface. 
 They are called *minimal*, and they can all be represented as real parts of some homomorphic $1$-forms. Minimal 1-forms on a surface $S$ are characterised by the property that for each point $x\in S$, where the one-form is non-vanishing there exists a circle $S^1$ on $S$ such that the one-form restricted to $S^1$ has not zeros while $x\in S^1$.
 
For example, in the case of a $T^2$ a one-form is harmonic for some metric iff it has no zeros at all.