Consider the image below where we have two point masses $m_1$ and $m_2$ with different masses which are rotating around a fixed axis with angular velocity $\omega$. If the origin is placed on the axis between the masses (left image), then angular momentum vectors $L = r\times p$ are parallel with the axis and does not change upon rotation. Hence torque is zero. If we move the origin away from the line between the masses, the angular momentum of each particle does not lie along the rotation axis and the total momentum will neither lie along the rotation axis. Therefor there must be a torque present to change the angular momentum during rotation. So in one case there is a torque and in the other there isn't???
Of course the two systems are physically the same and the same forces must be acting so there must be an error in the reasoning. Where is it?