Carbon nanotubes’ resilience to mechanical deformation is a potentially important feature for imparting tunable properties at the nanoscale. The influence of mechanical deformation on the thermal transport of carbon nanotubes is studied by non-equilibrium molecular dynamics. Nanotubes of different bending angles, lengths, diameters, chiralities, and degrees of twist are simulated in the regime in which the thermal transport extends from ballistic to diffusive. The study in purely bent carbon nanotubes settles the controversy around the differences between the current experimental and molecular dynamics measurements of the thermal transport in bent nanotubes. Collapsed carbon nanotubes, in contrast with graphene nanoribbons, which are known to exhibit substantial rough-edge and cross-plain phonon scatterings, preserve the quasiballistic phononic transport encountered in cylindrical nanotubes. Stacked-collapsed nanotube architectures, closely related with the strain-induced aligned tubes occurring in stretched nanotube sheets, are shown to inherit the ultrahigh thermal conductivities of individual tubes and are therefore proposed to form highways for efficient heat transport in lightweight composite materials.