Yes, snowshoeing requires a trail pass.
A trail day pass must be purchased to be on any of the trails at Tahoe Donner Cross Country Ski Center (TDXC). You also need to have snowshoes or skis on to use the trails. You can also purchase a season pass with your ShopTD account or onsite at TDXC.
You are welcome to use the Tahoe Donner trail system outside the Cross Country Center for snowshoeing (not groomed). There is no hiking allowed on the Cross Country trails after they open for the season.