Despite the increasing interest in the physical properties of the newly synthesized three-dimensional (3D) nano-architectured graphene, there are still few studies on the thermal transport properties of this family of materials. In the present work, heat transport of 3D h-BN and its mechanical response are systematically explored through first principles calculations. It is fascinating to find that the thermal conductivity of the 3D h-BN honeycomb structure could be significantly modulated by mechanical tension. Its lattice thermal conductivity perpendicular to the hole axis increases by 7.2 times at 6% critical strain, compared to only 0.67 times for that of the strained 3D graphene counterpart. The structure's thermal conductivity versus mechanical tension differs quantitatively and qualitatively from the monotonic downward trend of traditional bulk diamond or silicon under tension. This deviation from the classic behavior could be attributed to the modification of the phonon lifetimes, together with the competition between group velocities of low- and high-lying phonons under strain. Finally, the phonon vibrational modes contribution analysis indicates that the BN ribbon atoms contribute mainly at a lower frequency range. Our results provide important insights into potential employment of nanoarchitectured 3D white graphene for thermal management in relevant industrial applications.