Are there any materials that do not heat up linearly with time when exposed to a constant heat source?

Answer 1

the mechanism of heat transfer matters. For radiation it varies like T^4 , for conduction (Ts-T) and for convection ???
I assume you mean by conduction but the hotter the material gets the smaller the temperature difference and the slower the transfer of heat.
Even if Ts >>T so the % change in temperature is small, you would need the specific heat to be constant with temperature, which in general it is not except over limited temperature range for most materials.
So the answer is, linear behavior is in general not the case but can be used as an approximation in special cases.