An appliance for washing objects is disclosed that effectively and energy-efficiently filters oil-based residue from a fluid such as washing fluid that drains from a basin in the appliance. In one illustrative embodiment, an appliance includes a novel filter assembly for filtering oil-based residue out of the washing fluid. The filter assembly includes a filter layer composed of a material that is both oleophilic and hydrophobic. The filter assembly forms part of a fluid distribution system that has different configurations. A first configuration allows for recirculating filtered wash fluid from an outlet of the filter assembly back into the basin of the appliance. A second configuration allows for draining both the washing fluid from the outlet of the filter assembly and the oil-based residue from a bypass zone of the filter assembly to an appliance outlet.