Does gravity affect the time evolution of a QM wave function?

There is no theory of quantum gravity yet, but we can say that also in quantum mechanics, gravitational time dilation is affecting mass particle quantum systems. This fact is already used in quantum physics: The measured time (of the laboratory clock) is the time after gravitational time dilation (redshifted with respect to proper time), and from this measured time may be retrieved the proper time of the quantum system if we know the gravity forces which are acting on it.

If you're talking about the non relativistic Schrodinger equation $$ i \hbar \frac{d}{dt} \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi + V(x) \psi $$ then a gravitaional field effects the particle by changing the external potential $V(x)$ to be the gravitational potential energy at position $x$. In this case, no, there is no time dilation present.

If you want a relativistic theory of quantum mechanics, then you have to use quantum field theory. You could couple a quantum field to a fixed background metric $g_{\mu \nu}$, and in that case yes, there would actually be gravitational time dilation. The "wave function" of a particle is not perfectly well defined in curved space time, but I would say yes, for most intents you could say that the "wave function would spread slower" higher in a gravitational field.