Some properties of nanoscale multilayers, as for example the giant magnetoresistance, are expected to depend sensitively on the structure of the interphase boundaries. Recent experimental work aimed to elucidate the effect of annealing on the multilayer structure and properties. In the present study, thermally induced changes of coherent phase boundaries in multilayers with largely immiscible components were investigated by means of the Monte Carlo method based on a vacancy migration algorithm. Two limiting cases of the as-deposited state were considered: chemically sharp, ideally planar phase boundaries as well as strongly mixed, diffuse interfaces. Initially, a rapid demixing and sharpening of concentration profiles were observed. The morphological roughness of chemically nearly sharp interfaces was found to increase in the course of annealing at higher temperatures. The analysis of the evolving phase boundary topography suggests that the observed instability of thin layers of a few monolayers thickness is due to the development of a long-wavelength roughness. The Monte Carlo studies are compared with predictions of an analytical theory on surface roughening.