Your best bet might be something out of one of the SNO-PARKS. They are state-run so won't be affected. Make sure you pick up a pass and stick it in your window.
http://ohv.parks.ca.gov/?page_id=24867Lake Alpine is a nice destination, an easy hike from the car, nice camping, possibly with running water. Some of the little peaks around there make nice hikes. I don't think any additional permits are necessary.
Carson Pass is also fun, there's Round Top and a bunch of other peaks around there. Pick up a permit on the way, just past Pioneer, free, though I'm not sure they will be open (just a formality - there's never anyone up there this time of year to bother you).
http://www.fs.fed.us/r5/eldorado/recrea ... ke/permit/And there's lots of stuff around Tahoe. Even if all the land surrounding the SNO-PARKS might be federal, I don't think anyone will really consider it "closed."