The simple answer is, not necessarily. Not every piezo motor requires a PID control loop. However, most piezo motors operate closed-loop with some type of feedback, and PID control is only one possibility, but it is common and effective in controlling piezo motors. Some piezo motor applications can and do run open loop, although piezo actuators (different from piezo motors) are more likely to do so.
Piezo motors have a number of benefits including their small size, high-resolution capability, and high torque developed relative to their small size. They also have low inertia, which makes them more responsive with fast start and stop times. For optimal performance (meaning better accuracy and repeatability) a closed-loop control scheme is usually a part of a piezo motor system. Common feedback devices that provide input for controllers can include capacitive sensors, laser interferometers, as well as optical incremental encoders, depending on the application.
A typical controller for a piezo motor can be a standard off-the-shelf servo controller available from any number of manufacturers, or it can be a specialized PID controller. Another type of control offered by some manufacturers is the so-called space state controller. In this type of controller, the resonance frequency is dampened leading to faster settling times.