Is there a quantum analogue of the "Tennis Racket" theorem?

The question is whether the quantum rigid rotor without axial symmetry shows any remnant of the intermediate axis ("tennis racket") theorem. We would expect that in the semi-classical limit (large excitation energy) the answer is yes.

The problem has indeed been studied in nuclear and molecular physics. A nice discussion can be found in Sect 4.5 of volume II of Bohr & Motelson, Nuclear Structure.

At low excitation energy there is no remnant of the intermediate axis theorem. The spectrum is discrete, and the states can only be labeled by discrete symmetries (there is no continuous symmetry, we cannot simultaneously diagonalize any $L_i$ and $H$, and there is no sense in which the system is spinning around a fixed axis).

In the semi-classical limit we are looking for Regge trajectories, approximately continuous energy eigenavlues that approach the classical energy. If there is a conserved $L_i$ it is clear how to do that. The spectrum is of the form $l(l+1)$, which is analytic in $l$ and approaches the classical result $l^2$. Without a conserved $L_i$ it is not entirely clear how to proceed. B & M suggest the following procedure. Work in a basis in which the eigenvalue $I$ of $L_i$ along a given principal axis is fixed. Study the Hamiltonian in the $I$ subspace, and find the analytic dependence of the energies on $I$. For the largest and smallest moment of inertia axes this leads to the expected spectrum. For the intermediate axes quasi-classical rotational states do not exist. This makes intuitive sense to me: If I try to initialize the system in an intermediate axis quasi-classical state, I get a linear combination of minimum/maximum axis states, which will lead to a quasi-classical version of tumbling.

Thomas is correct; the tennis racket theorem can't be proved in quantum mechanics.

So, Euler's equations say that without a torque being applied to an object, the rate of change of the angular momentum around one principal axis of a body is directly proportional to the product of the angular momentum around the other two axes. One proves the intermediate axis theorem by first stating that the angular momentum around one axis is much larger than the other two, and then goes to work. This doesn't fly in quantum mechanics; as you probably know, $$mathbf{L}timesmathbf{L}=ihbarmathbf{L}$$

Thus, there isn't a basis where you can find a state where all three (or even just two) are simultaneously well-defined, so you can't really perform the analysis necessary. You can construct quantum Euler equations, (see Casimir's Rotation of a Rigid Body in Quantum Mechanics), but you can't prove the tennis racket theorem from them.