We study the thermal transport properties of twisted-layer graphite nanofibers. We show that in the presence of a twisted layer, the phonon thermal conductance of a graphite nanofiber of rectangular cross-section varies remarkably with the twist angle and can reach minimum values either at two critical angles θ1 and θ2 that conform to the rule θ1 + θ2 = 1800 or exactly at the angle θ = 900. A reduction of roughly 50% of the phonon thermal conductance can be achieved in some structures. We unveil that the twisting effect mainly influences the optical modes, more precisely, the breathing modes ZOꞌ, and leaving almost unaltered the acoustic ones. The effect is also visible in the higher and more numerous van Hove singularities of the phonon density of states. We also point out that the behavior of the thermal conductance with the twist angle is associated with and dominated by the alteration in the overlap area between the wisted and non-twisted layers. The finite-size effect is demonstrated to play an essential role in defining the critical angles at the local minima, where these angles are dependent on the size of the investigated nanofibers, in particular on the proportion between the widths of zigzag and armchair edges. We also analyze the behavior of circular nanofibers where the correlation between the overlap area and the thermal conductance is much smaller.