Transport measurements have been an indispensable tool in studying conducting states of matter. However, there exists a large set of interesting states that are insulating, often due to electronic interactions or topology, and are difficult to probe via transport. Here, through an experiment on carbon nanotubes, we present a new approach capable of measuring insulating electronic states through their back action on nanomechanical motion. We use a mechanical pump-probe scheme, allowing the detection of shifts in both frequency and dissipation rate of mechanical vibrational modes, in an overall insulating system. As an example, we use this method to probe the non-conducting configurations of a double quantum dot, allowing us to observe the theoretically predicted signature of nanomechanical back action resulting from a coherently tunnelling electron. The technique opens a new way for measuring the internal electronic structure of a growing variety of insulating states in one-and two-dimensional systems.