Do any of our satellites or space exploration craft need or use relativity for strictly navigational purposes?
Keep in mind that interplanetary probes don't know where they are in space until they get close to the target object (and sometimes not even then). The onboard computer of a vehicle about to land on Mars doesn't need to consider third body perturbations, let alone relativistic effects. For example, the Mars Science Laboratory Entry, Descent, and Landing flight software modeled gravity using a simple J2 (equatorial bulge) gravity model of Mars. This simple model was the largest contributor to the error in the landed position; Mars exhibits significant gravitational anomalies. The errors that resulted from this simplistic model of Mars gravity field are orders of magnitude than are the errors that resulted from ignoring the Newtonian gravitational effects of the Sun and Jupiter, which in turn are orders of magnitude larger than the errors that resulted from ignoring the relativistic effects.
Navigation of a spacecraft in interplanetary space is performed by people and computers on the Earth rather than by the spacecraft. Earth-bound computers don't have the huge constraints imposed on onboard computers, so they can model relativistic effects. Those relativistic effects are important for a spacecraft that takes many months to many years to get from Earth to the target. As a result, the Deep Space Network uses a Parameterized Post-Newtonian (PPN) formulation to calculate the effects of gravitation on spacecraft; see section 4 of Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation.