Most of the time, no. Even in the dead of winter, socks get too sweaty and uncomfortable under the covers.
I do have a set of fuzzy grippy socks that seem to breathe much better than ordinary socks. I guess it's similar to "fleece" material, though the texture is a bit different? They're simultaneously warmer than my regular socks, and yet more comfortable to wear in bed. Unfortunately they don't look much like "real" socks, nor do they fit well in all of my shoes, so I can't just replace all my regular socks with these.