Thank you for asking this very important question! First thing to know is that all parts of the Nandina plant (roots, leaves, etc.) carry the toxin, which can be absorbed by your skin whilst you're doing your work on the plant, so please be sure to gear up appropriately.  Removing the berries would be an excellent first step (put in garbage, not yard recycling, please) because with the freeze-thaw cycle and natural foods being less available, birds will be drawn to consuming the berries. Removing the plants will be a bit of a prolonged project, I'm afraid. Another Extension pro provided some information that will prove useful:

Depending on the width and location, the first step would be to cut them back so you can get around it to dig out. Then just try to dig as much of the root mass as possible. The roots can be extensive. If they are in a location that is easier to get to and you can drive a vehicle close by you can attempt to wrap a strong rope or chain around the branches and rip them out. (do this slowly and with extreme caution, make sure no one is near by to get hit.) But you will be compacting the soil and have potential of harming other plant that you may want to keep, so this scenario really only works if its accessible.

Otherwise once the plants are out of the ground then you'll just want to monitor any regrowth from berries or roots left in the soil and dig out as they come.

If they are in a location or you are unable to dig or pull them out, you can cut them back to the ground as closely as possible and use a systemic herbicide applied to the cuts to help kill the roots. Again you would just do this repeatedly until the plant fails to send up any new growth.