The hive, from bottom to top, is a stand (something that sits on the ground that the bottom board goes on) a bottom board, some boxes with frames (brood boxes or supers depending on their use), some kind of cover (sometimes with an inner cover under it), some bricks or rocks to hold the lid on.
If you have a Screened Bottom Board there may be a tray that slides under it.
So if you mean the bottom board, yes, it stays on all year. If you mean the tray, some people leave it out except when they want to monitor Varroa mite drops and others put it in over the winter. I put it in over the winter.