What would be the consequence of the inductor saturating?
One thing that happens as the core begins to saturate is that the inductance value drops, precipitously at saturation. This will change the filters AC characteristics. In a filter application, obviously the AC signals you are trying to block becomes less blocked. Many specifications for inductors intended for filter applications will provide inductance vs DC current. This is a function of the core material. For all practical purposes air can't be saturated, but as far as I know all ferro-magnetic materials can be.