The answer depends on the nature of the force that is causing the simple harmonic motion (the "restoring force").

In the case of a simple pendulum, the restoring force is the weight of the object. This is proportional to the mass of the object. So if the mass of the object changes, the restoring force changes by the same proportion, and the period of the pendulum does not change.

However, in the case of an object oscillating on a spring, the restoring force is determined by the spring constant $k$, which is independent of the mass of the object. If the mass of the object changes then the restoring force does not change by the same proportion, so the period of oscillation does change as the mass of the object changes. Once you have a changing period then the motion is no longer simple harmonic motion.

The answer depends on the nature of the force that is causing the simple harmonic motion (the "restoring force").

In the case of a simple pendulum, the restoring force is the weight of the object. This is proportional to the mass of the object. So if the mass of the object changes, the restoring force changes by the same proportion, and the period of the pendulum does not change (if we assume a point mass for simplicity).

However, in the case of an object oscillating on a spring, the restoring force is determined by the spring constant $k$, which is independent of the mass of the object. If the mass of the object changes then the restoring force does not change by the same proportion, so the period of oscillation does change as the mass of the object changes. Once you have a changing period then the motion is no longer simple harmonic motion.