What is the necessary condition on a force the result the conservation of angular momentum for a particle affected by that force?

The necessary condition is that the force that is applied should not be able to produce any torque about the axis of rotation which is achieved only if the force passes through the axis of rotation.

Explanation:

Mathematically the angular momentum is related to force as

where

'r' is arm of the force 'F'

For angular momentum to be conserved the right hand side of the above equation should be zero which can happen only if:

1) F = 0

2) r = 0

3) F and r are parallel.

Thus fulfilling any of the above condition shall conserve the angular momentum of a particle.