Can you calculate the static lattice/potential energy given a phonon dispersion curve? Let's say I determine a material's dispersion relation from experiments. Would it then be possible to use the information contained within the dispersion relation to calculate the material's static lattice energy? 
 A: No, it is impossible. The reason is very simple if one recalls the meaning of phonon dispersion curves. They give the set of possible frequencies, $\omega_s({\bf k})$, for the eigenmodes of the $s-$th dispersion branch. 
The energy density of a harmonic crystal, at a temperature $T$ is:
$$
u = u_{static} + \frac{1}{V}\sum_{{\bf k},s}  \frac{1}{2} \hbar  \omega_s({\bf k}) + \frac{1}{V}\sum_{{\bf k},s}  \frac{ \hbar  \omega_s({\bf k})}{e^{ \frac{ \hbar \omega_s({\bf k})}{k_B T}}-1},
$$
where the second term is independent on $T$ and corresponds to the so-called zero-point contribution. $u_{static}$ is the energy of a (static) perfect lattice.
It is clear that the whole information on phonons is contained in the second and third terms. The first term is the (static) potential energy of a set of particles at the equilibrium positions in the lattice. Even though both $u_{static}$ and $ \omega_s({\bf k}) $ are obtained from the same potential energy as a function of the atomic positions, they contain different information and one cannot obtain one of them from knowledge of the other.
