To date, there has been limited reporting on the fabrication and properties of macroscopic sheet assemblies (specifically buckypapers) composed of carbon/boron nitride core–shell heteronanotubes (MWCNT@BNNT) or boron nitride nanotubes (BNNTs). Herein we report the synthesis of MWCNT@BNNTs via a facile method involving Atmospheric Pressure Chemical Vapor Deposition (APCVD) and the safe h-BN precursor ammonia borane. These MWCNT@BNNTs were used as sacrificial templates for BNNT synthesis by thermal oxidation of the core carbon. Buckypaper fabrication was facilitated by facile sonication and filtration steps. To test the thermal conductivity properties of these new buckypapers, in the interest of thermal management applications, we have developed a novel technique of advanced scanning thermal microscopy (SThM) that we call piercing SThM (pSThM). Our measurements show a 14% increase in thermal conductivity of the MWCNT@BNNT buckypaper relative to a control multiwalled carbon nanotube (MWCNT) buckypaper. Meanwhile, our BNNT buckypaper exhibited approximately half the thermal conductivity of the MWCNT control, which we attribute to the turbostratic quality of our BNNTs. To the best of our knowledge, this work achieves the first thermal conductivity measurement of a MWCNT@BNNT buckypaper and of a BNNT buckypaper composed of BNNTs not synthesized by high energy techniques.