Super-Earths, planets with masses between Earth and Neptune, are a new, exciting subject of astrobiology as some of them could be potentially habitable—particularly within planetary systems orbiting M dwarf stars, the most common stars in the Milky Way. However, whether interior properties of super-Earths are capable of fostering plate tectonics remains unconstrained but is critically important because plate tectonics plays a significant role in Earth's surface conditions necessary for the emergence of habitats suitable for complex life. We explore interior dynamics of ten representative super-Earths that were previously suggested as potentially geophysically benign and habitable. Through our thermochemical calculations of mantle melting and surface volcanism in the super-Earths, we found that five may not experience mantle melting, and another one may undergo volcanism but without delivering water to its shallow oceans on account of excessive water solubility during solidification of magma beneath seafloor. Our result suggests that even with conservative parameterizations, plate tectonics is inhibited across multiple candidate habitable super-Earths as a critical prerequisite for sustaining long-term habitats suitable for complex life as found on Earth. This result indicates the possibility that habitable conditions on rocky extrasolar planets may only arise with certain planetary parameters conducive to melting in both their rocky mantles and icy layers—making potentially habitable conditions relatively uncommon around small red-dwarf stars—unless there exist as-yet-unexplored mechanisms responsible for melting in rocky mantles of potentially habitable super-Earths.