This numerical study examines the flow behavior of a viscoplastic fluid governed by the Herschel-Bulkley rheological model within a pipe subjected to a constant wall temperature. Both the shear-thickening fluid and the limiting case of a Bingham fluid are considered. The governing equations are solved using an in-house computational code based on the finite volume method. Particular attention is given to the variation of the friction factor, and consequently the pressure drop, under two conditions: assuming an isoviscous fluid and a temperature-dependent one, both with the inclusion of viscous dissipation. The results indicate that neglecting the temperature dependence of the fluid’s consistency, as well as the effects of viscous dissipation, results in an underestimation of the pressure drop. Such simplifications may lead to significant errors in the design of industrial systems. Furthermore, the results show that wall heating yields the lowest friction factor values, although these values remain higher in the case of isoviscous flow.