It has long been known that weak electron temperature gradients in fully ionized plamas (satisfying λ ∣∇Te∣/Te ≲ 10-4, where λe is the electron mean free path and Te is the electron temperature) can lead to the development of significant non-Maxwellian suprathermal tails on electron velocity distributions, invalidating the Spitzer and H¿rm <1953> perturbation theory <Gray and Kilkenny, 1980; Bell et al., 1981; Scudder and Olbert, 1983>. In this paper we work out the implications of such nonlocal heat flow for electrons in the solar corona, comparing a new analytical theory to numerical solutions of the Fokker-Planck equation. While electron-electron Coulomb collisions are strong enough at coronal densities to influence the local temperature, the electron heat flux is determined by the essentially collisionless high-energy tail. The deceleration of suprathermal electrons in the polarization electric field allows electron heat to flow radially outward against the local temperature gradient, in contrast to the local thermodynamic equilibrium picture, in which heat is constrained to flow down the local temperature gradient. We discuss the implications of this effect for empirical constraints of coronal heating mechanisms.