Generally the road "closes" after the first snow. However, the barricade is easily passed and usually is. I've seen two closure points, the first is at the Lone Pine Campground turn off; the second at where the road starts to switchback up the mountain.
Inyo County doesn't plow the road and there usually a ton of debris stewn across, so you have to run a gaunlet of boulders to get further.
We have been up there the Mid-March the last two years and got to the Meysen Lake Trailhead parking last year and to the Portal Store the year before. How far you get is dependent on the amount of snow and debris.
Bill