Is there an example of a closed shell molecule with a permanent ground state magnetic dipole?
Updated 4/6/2017: It turns out that singlet ($^1\Delta_g$) molecular oxygen is paramagnetic due to orbital angular momentum only, as all electrons are paired. It has a magnetic dipole moment of $1\hbar$ and can be determined experimentally. (This makes sense from a computational perspective, given that the lowest restricted Hartree Fock solution of singlet oxygen is always complex, and only complex wave functions yield non-zero expectation values of orbital angular momentum.) 



For closed shell (even number of electrons) molecules the answer is no.

After a little more reading, I found the answer. The reason is that the magnetic dipole is odd under time reversal (Table 4.1, p. 218 in Barron, 2004).

Apparently, (Barron, p. 221) "the expectation value of a time-odd operator [e.g. magnetic dipole] vanishes for states invariant under time reversal, which can always be constructed for an even electron system and hence any nondegenerate state".

I do not believe this applies to transition properties, however, as we can see magnetic dipole allowed transitions.

Citation: Barron, Laurence D. Molecular light scattering and optical activity. Cambridge University Press, 2004.