In the fabrication of optical fibers, the viscosity of the glass varies dramatically with temperature so that heat transfer plays an important role in the deformation of the fiber geometry. To date, models of fiber drawing processes have either assumed isothermal conditions or that advection is the dominant heat-transfer mechanism, with conduction therefore neglected. While advective heat transfer and neglect of conduction are valid in the fast process of drawing a preform to a fiber, heat conduction is also important in the slow process of extruding a fiber preform. In this paper, we derive and solve, in the context of pulled extrusion, a coupled heat and fluid flow model that incorporates heat conduction. The dramatic variation of viscosity with temperature means that this problem is extremely challenging to solve via standard numerical techniques and we therefore develop a novel finite-difference numerical solution method that proves to be highly robust. We use this method to show that conduction significantly affects the size of internal holes at the exit of the device and hence has important consequences for the process of manufacturing optical fibers.