We demonstrate theoretically that the resonance frequencies of high-Q microcavities in two-dimensional photonic crystal membranes can be tuned over a wide range by introducing a subwavelength dielectric tip into the cavity mode. Three-dimensional finite-difference time-domain simulations show that by varying the lateral and vertical positions of the tip, it is possible to tune the resonator frequency without lowering the quality factor. Excellent agreement with a perturbative theory is obtained, showing that the tuning range is limited by the ratio of the cavity mode volume to the effective polarizability of the nanoperturber.