We report on the studies of crystallographic tilt induced by miscut of the Si (111) substrate in GaN-on-Si (111) heterostructures grown by metal-organic chemical vapor deposition. By employing high-resolution X-ray diffraction, we found that the onset of crystallographic tilt occurred at the interface between the AlN nucleation layer and the Si (111) substrate. The orientation of the GaN overlayer always follows that of the AlN nucleation layer irrespective of its quality and miscut of the substrates. The resultant GaN [0002] is tilted toward GaN (11-20) and (10-10) atomic planes for the miscuts of Si (111) toward Si [1-10] and [11-2], respectively. In both cases, the misorientation of GaN (0002), i.e., the tilt of GaN [0002] from the surface normal direction, is in the same direction of the miscut of Si (111). The misorientation angle of the GaN epilayer is generally smaller than the miscut angle of the substrate. However, the crystallographic tilt, i.e., the angle formed between GaN [0002] and Si [111], is always much larger than the Nagai tilt. These observations are attributable to misfit dislocations that are anisotropically generated at the AlN/Si (111) interface. This mechanism is discussed based on recent microscopic observations of in-plane misfit dislocations at the interface near the atomic step edges.