The effect of high pressure on the thermal conductivity of bundles of single- and double-walled carbon nanotubes and C70 fullerene is studied. The pressure regions in which the thermoelectric properties of these structures significantly change are determined. It is found that the pressure dependences are indicative of the partially reversible structural transformations of carbon nanotubes and the irreversible amorphization of C70 fullerene at a pressure of ~50 GPa.