Membrane nanotubes are continuously assembled and disassembled by the cell to generate and dispatch transport vesicles, for instance, in endocytosis. While these processes crucially involve the ill-understood local mechanics of the nanotube, existing micromanipulation assays only give access to its global mechanical properties. Here we develop a new platform to study this local mechanics using atomic force microscopy (AFM). On a single coverslip we quickly generate millions of substrate-bound nanotubes, out of which dozens can be imaged by AFM in a single experiment. A full theoretical description of the AFM tip-membrane interaction allows us to accurately relate AFM measurements of the nanotube heights, widths, and rigidities to the membrane bending rigidity and tension, thus demonstrating our assay as an accurate probe of nanotube mechanics. We reveal a universal relationship between nanotube height and rigidity, which is unaffected by the specific conditions of attachment to the substrate. Moreover, we show that the parabolic shape of force-displacement curves results from thermal fluctuations of the membrane that collides intermittently with the AFM tip. We also show that membrane nanotubes can exhibit high resilience against extreme lateral compression. Finally, we mimic in vivo actin polymerization on nanotubes and use AFM to assess the induced changes in nanotube physical properties. Our assay may help unravel the local mechanics of membrane-protein interactions, including membrane remodeling in nanotube scission and vesicle formation.