Magnetosensitive elastomers respond to external magnetic fields by changing their stiffness and shape. These effects result from interactions among magnetized inclusions that are embedded within an elastic matrix. Strong external magnetic fields induce internal restructuring, for example the formation of chain-like aggregates. However, such reconfigurations affect not only the overall mechanical properties of the elastomers but also the transport through such systems. We concentrate here on the transport of heat, that is thermal conductivity. For flat, thin model systems representing thin films or membranes and modeled by bead-spring discretizations, we evaluate the internal restructuring in response to magnetization of the particles. For each resulting configuration, we evaluate the associated thermal conductivity. We analyze the changes in heat transport as a function of the strength of magnetization, particle number, density of magnetizable particles (at fixed overall particle number), and aspect ratio of the system. We observe that varying any one of these parameters can induce pronounced changes in the bulk thermal conductivity. Our results motivate future experimental and theoretical studies of systems with magnetically tunable thermal but also electric conductivity---both of which have only rarely been addressed so far.