The production of continuous carbon nanotubes (CNT) fibers and films has paved the way to leverage the superior properties of individual carbon nanotubes for novel macroscale applications such as electronic cables and multi-functional composites. In this manuscript, we synthesize fibers and films from CNT aerogels that are continuously grown by floating catalyst chemical vapor deposition (FCCVD) and measure thermal conductivity and natural convective heat transfer coefficient from the fiber. To probe the mechanisms of heat transfer, we develop a new, robust, steady-state thermal characterization technique that enables measurement of the intrinsic fiber thermal conductivity and the convective heat transfer coefficient from the fiber to the surrounding air. The thermal conductivity of the as-prepared fiber ranges from 4.7±0.3 W m^-1 K^-1 to 28.0±2.4 W m ^-1 K ^-1 and depends on fiber volume fraction and diameter. A simple nitric acid treatment increases the thermal conductivity by as much as a factor of ~3 for the fibers and ~6.7 for the thin films. These acid-treated CNT materials demonstrate specific thermal conductivities significantly higher than common metals with the same absolute thermal conductivity, which means they are comparatively light weight, thermally conductive fibers and films. Beyond thermal conductivity, the acid treatment enhances electrical conductivity by a factor of ~2.3. Further, the measured convective heat transfer coefficients range from 25 W m^-2 K^-1 to 200 W m^-2 K^-1 for all fibers, which is higher than expected for macroscale materials and demonstrates the impact of the nanoscale CNT features on convective heat losses from the fibers and films. The measured thermal and electrical performance demonstrates the promise for using these fibers and films in macroscale applications requiring effective heat dissipation.