We study modulation instability (MI) of random-phase waves in nonlinear photonic lattices. We find that an incoherent superposition of extended nonlinear eigenstates of the system, that is, an incoherent extended stationary beam, may become unstable due to nonlinearity. The instability process depends on the nonlinearity, on the structure of the diffraction curves of the lattice, as well as on the properties of the beam, whose spectrum can be comprised of Bloch modes from different bands, and from different regions of diffraction (normal/anomalous). This interplay among diffraction, incoherence, and nonlinearity leads to a variety of phenomena, including the possibility of tailoring the diffraction curve of the lattice, or the coherence properties of the beam, to enhance or suppress the instability. We present several examples of such phenomena, including a case where increasing the lattice depth flattens the diffraction curve thereby enhancing the instability, ”locking” the most unstable mode to the edge of the 1st Brillouin zone for large nonlinearity, and incoherent MI in self-defocusing media.