Emerging studies at the nanoscale on the relationships between the structure, mechanical properties and infectivity of virus particles are revealing important physics-based foundations of virus biology that may have biomedical and nanotechnological applications. Human rhinovirus (HRV) is the major causative agent of common colds leading to important economic losses, and is also associated with more severe diseases. There is renewed interest in developing effective anti-HRV drugs, but none have been approved so far. We have chosen HRV to explore a possible link between virus mechanics and infectivity and the antiviral effect of certain drugs. In particular, we have investigated a suggestion that the antiviral action of drugs that bind to capsid cavities (pockets) may be related to changes in virus stiffness. Mechanical analysis using atomic force microscopy shows that filling the pockets with drugs or genetically introducing bulkier amino acid side chains into the pockets stiffen HRV virions to different extents. Drug-mediated stiffening affected some regions distant from the pockets and involved in genome uncoating, and may be caused by a subtle structural rearrangement of the virus particle. The results also revealed for HRV a quantitative, logarithmic relationship between mechanical stiffening, achieved either by drug binding or introducing bulkier amino acid side chains into the pockets, and reduced infectivity. From a fundamental physics perspective, these drugs may exert their biological effect by decreasing the deformability of the virion, thus impairing its equilibrium dynamics. The results encourage the design of novel antiviral drugs that inhibit infection by mechanically stiffening the viral particles.