I'm arguing here in a different way than the answer by @josephh. In my opinion, you can definitely apply the Schrödinger equation -- resp. a probably differently named version of it -- to phonons as well, but you have to extend the framework.
In the standard setting you're basing the whole formalism on a Hilbert space $\mathcal H_N$ with a definite particle number $N$, and by solving the SE you're searching for a solution $|\Psi_N\rangle \in \mathcal H_N$.
In the case of phonons, instead you have to use the Fock space $\mathcal F$, which is the direct sum of all Hilbert spaces with varying particle number $N$. In this space, you can define the physical processes of creation and annihilation in the sense that a wavefunction goes from subspace $\mathcal H_N$ to $\mathcal H_{N\pm1}$.
In practice, however, this detailed description is usually to complex, and so, as mentioned in the other answer, the density matrix is commonly used. It doesn't care about the microstate of your system, but rather gives the probability (density) of finding the particle at a point in space $x$.
By the way: In the same manner I'd also argue that you can apply the Schrödinger equation to a system with spin $\neq 0$ (even if it's called differently). The idea is similar: e.g. for electrons, instead of using the Hilbert space $\mathcal H$, one uses an extended (Hilbert) space $\mathcal H \times \{\uparrow,\downarrow\}$, where $\uparrow,\downarrow$ are spin-states of a spin-1/2 particle, and applies the (form of the) Schrödinger equation again, with a specific Hamiltonian. The particular form of the Hamiltonian then basically determines the kind of equation and its name (e.g. Pauli equation, etc.).