No, but there is an inertial torque you have to worry about.
From the perspective of an inertial frame, the rotational analog of Newton's second law for rotation about the center of mass is
where L is the object's angular momentum with respect to inertial, τ,i is the i external torque, and the differentiation is from the perspective of the inertial frame. Note that this pertains to non-rigid objects as well as rigid bodies.
The relationship between the time derivatives of any vector quantity q from the perspectives of co-located inertial and rotating frames is
where is the frame rotation rate with respect to inertial.
For a rigid body, the body's angular momentum with respect to inertial but expressed in body-fixed coordinates is where I is the body's moment of inertia tensor and is the body's rotation rate with respect to inertial but expressed in body-fixed coordinates. Since a rigid body's inertia tensor is constant in the body-fixed frame, we have
Combining equations (1), (2), and (3) yields
This is Euler's equations of motion for a rigid body. No inertial forces come into play. However, the term is essentially an inertial torque. Just as inertial forces vanish in inertial frames, so does this inertial torque.
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