Of the 14 transiting extrasolar planetary systems for which radii have been measured, at least three appear to be considerably larger than theoretical estimates suggest. It has been proposed by Bodenheimer, Lin & Mardling that undetected companions acting to excite the orbital eccentricity are responsible for these oversized planets, as they find new equilibrium radii in response to being tidally heated. In the case of HD 209458, this hypothesis has been rejected by some authors because there is no sign of such a companion at the 5 ms−1 level, and because it is difficult to say conclusively that the eccentricity is non-zero. Transit timing analysis as well as a direct transit search has further constrained the existence of very short period companions, especially in resonant orbits. Whether or not a companion is responsible for the large radius of HD 209458b, almost certainly some short-period systems have companions which force their eccentricities to non-zero values. This paper is dedicated to quantifying this effect. The eccentricity of a short-period planet will only be excited as long as its (non-resonant) companion's eccentricity is non-zero. Here, we show that the latter decays on a time-scale which depends on the structure of the interior planet, a time-scale which is often shorter than the lifetime of the system. This includes Earth-mass planets in the habitable zones of some stars. We determine which configurations are capable of sustaining significant eccentricity for at least the age of the system, and show that these include systems with companion masses as low as a fraction of an Earth mass. The orbital parameters of such companions are consistent with recent calculations which show that the migration process can induce the formation of low-mass planets external to the orbits of hot Jupiters. Systems with inflated planets are therefore good targets in the search for terrestrial planets.