The coalescence of a binary neutron star (NS) system (a 'NS merger' or NSM) may in some cases produce a massive NS remnant that is long-lived and, potentially, indefinitely stable to gravitational collapse. Such a remnant has been proposed as an explanation for the late X-ray emission observed following some short duration gamma-ray bursts (GRBs) and as possible electromagnetic counterparts to the gravitational wave chirp. A stable NS merger remnant necessarily possesses a large rotational energy ∼ > 10 52 erg, the majority of which is ultimately deposited into the surrounding circumburst medium (CBM) at mildly relativistic velocities. We present Very Large Array (VLA) radio observations of 7 short GRBs, some of which possessed temporally extended X-ray emission, on timescales of ∼ 1−3 years following the initial burst. No radio sources were detected, with typical upper limits ∼ 0.3 mJy at ν = 1.4 GHz. A basic model for the synchrotron emission from the blast wave is used to constrain the presence of a long-lived NSM remnant in each system. Depending on the GRB, our nondetections translate into upper limits on the CBM density n ∼ < 3 × 10 −2 − 3 cm −3 required for consistency with the remnant hypothesis. Our upper limits rule out a long-lived remnant in GRB 050724 and 060505, but cannot rule out such a remnant in other systems due to their lower inferred CMB densities based on afterglow modeling or the lack of such constraints.