Analogies between electrostatics and steady state heat equation?

Even though there is a certain similarity between the two equations, the origins are quite different. Maxwell's equations can be derived from a gauge theory, while the heat equation is a special case of a diffusion phenomenon (and hence a special case of the more general diffusion equation). Also note that the more general expression of the heat equation is

$$rho c_pfrac{partial T}{partial t} - nablacdot(kappanabla T) = dot q$$

where $rho$ is the mass density, $c_p$ is the specific heat capacity, $dot q$ is the volumetric heat source, and $kappa$ is, in general, a tensor. If we look for stationary solutions (i.e. $partial_tT = 0$), for a homogeneous and isotropic medium (i.e. $kappa$ is a space-wise constant scalar) then the equation becomes

$$nabla^2 T = -frac1kappadot q.$$

The RHS can still be interpreted as a density of heat source in space, much like $rho$ in the electrostatic equation is a density of electric charge (divided by $varepsilon$). However, for the heat equation, we have a time derivative, $dot q$, that alludes to a (heat) flux, which is not the case for the charge density $rho$. Indeed, the electrostatic equation is not describing a diffusion phenomenon.