We investigate anomalous ion-motional heating, a limitation to multiqubit quantum-logic gate fidelity in trapped-ion systems, as a function of ion-electrode separation. Using a multizone surface-electrode trap in which ions can be held at five discrete distances from the metal electrodes, we measure power-law dependencies of the electric-field noise experienced by the ion on the ion-electrode distance d. We find a scaling of approximately d−4 regardless of whether the electrodes are at room temperature or cryogenic temperature, despite the fact that the heating rates are approximately two orders of magnitude smaller in the latter case. Through auxiliary measurements using the application of noise to the electrodes, we rule out technical limitations to the measured heating rates and scalings. We also measure the frequency scaling of the inherent electric-field noise close to 1/f at both temperatures. These measurements eliminate from consideration anomalous-heating models which do not have a d−4 distance dependence, including several microscopic models of current interest.