[This is an answer made of two comments and an example]
Since the laplacian is elliptic with real-analytic coefficients, a harmonic function $f$ is real-analytic in its domain of definition. Hence the set $C$ of critical points of $f$ is a real-analytic subset of $R^3$, and as such it admits a locally finite partition into real-analytic locally closed smooth submanifolds. Thus if $\dim C≤1$, it is locally a finite union of analytic open arcs and singular points (but the curves might not extend smoothly across those points).
A reference on real analytic functions (reedited in 2002) might be
S. Krantz, H. Parks, A primer of real analytic functions. Birkhäuser Verlag, 1992.
But maybe the "curve selection lemma" in Milnor's "Singular points on complex hypersurfaces" would be enough.
As an example of a curve of critical points not extending throuh a singular point, take the harmonic polynomial $f(x,y,z)=y^3-3x^2y+y^3z-yz^3$, which has critical locus $y=0,z^3=-3x^2$.