Are the harmonic maps from a 2dimensional cone to $S^2$ locally lipschitz or Holder continuous?

They are localy Holder continuous. There is a map $\phi$ from the plane to the cone which is bijective, the inverse is Holder, and $\phi$ is conformal except one point. This map can be written explicitly. Let your map be $f$. Composition $g=f\circ\phi$ is harmonic except one point, so by removable singularity theorem it is harmonic everywhere, thus locally Holder. So your map $f=g\circ\phi^{1}$ is locally Holder. The Holder exponent depends on the opening of the cone. 

