Why is $\mu=0$ for p-dinitrobenzene ? How can the dipole moments of the two Nitro groups exactly cancel each other ? Shouldn't it have a non-zero, albeit feeble, dipole moment ?
This paper (J. Sep. Sci. 2011, 34, 1489–1502) lists 1,4-Dinitrobenzene as having a very small dipole moment (0.01 Debye). No error is given on the dipole moments, but they do list compounds with exactly zero dipole moment (naphthalene for example).
Let the two nitro groups have a different rotational excitation against the aryl core, and i'm sure you can construct a small dipole moment. Firstly, higher rotational states have different bond angles.
Or easier to imagine: Let one stand in plane with the aryl ring, and one perpendicular or rotating. Different interaction with the aromatic ring, clearly leads to slightly different electronic structure, ergo: a dipole moment.
The dipole moment should vanish at low temperatures, but i guess the stuff crystallises before.