Non stationary effects of molecular orbitals
First off, let's define our terms. The quasistatic approximation:


  More generally, the quasistatic approximation can be understood through the idea that the sources in the problem change sufficiently slowly that the system can be taken to be in equilibrium at all times.


So the question boils down to "are there ever any quantum mechanical behaviors that are non-equilibrium?" Or. more precisely, are there fast changes where the system is not really in equilibrium.

Well, as noted in the comments, there's the Born-Oppenheimer approximation which, at least in my opinion, reflects a quasi-static view of electronic structure. That is, the nuclei move slowly relative to the electrons, so despite vibrations, etc. the electrons can be considered in equilibrium with nuclear positions.

Obviously the Born-Oppenheimer approximation fails.

The poster suggested the question was about other non-static behavior. In that case, you fall into time-dependent quantum mechanics, e.g. time-dependent Hartree Fock (i.e., RPA) or TD-DFT.


  Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system. In this case one can analyze the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function.


Are there cases where linear response breaks and the variation in the system doesn't depend only on the ground-state wave-function. Yes. Consider the effects of large perturbations, like large electric fields, nonlinear optical effects (two photo absorption), etc.

As another fairly easy example, consider photo-ionization. That is, remove an electron from a system. In the quasi-static approximation, you'd have something like Koopmans' theorem with frozen ground-state orbitals. In reality, there's orbital relaxation even though the ionization process is very fast.