Lilium bulbs are a little different than bulbs that go completely dormant. By all means, remove the dead roots. But the live roots are needed for the following growing season. Lilium bulbs don't need to be stored in the fridge with soil in a pot, but they need to be in a closed bag, with just enough moisture to keep bulb and roots from dehydrating. Optimum temperature for Lilium bulbs is minus 1 to 4°C.
Because you mentioned in your description of the roots that they are like a wig, that many roots usually occur in Amaryllis, Habranthus and Lycoris species, rather than in Lilium. I just want to make sure we are talking about the correct plant.