I plan to store my model 3 during the winter months. Any precautions that I should consider? If I am not driving it, do I need to keep it on the charger? Will I need to occasionally “run it” to sustain the battery?

I don't see any reason why you should keep it permanently on the charger. Check status occasionally with your app, and adjust climate remotely to exercise the heating and ventilation. That will discharge the battery more than just the usual from just sitting there. When the battery falls to 20% plug it into 110V and let it slowly recharge up to 80%.

You should probably plug it in for the duration and set the charge limit to 70%, or so. Take it around the block every month or two just to keep the tires from flat-spotting. Other than that, you should be all set, i would think.