When I started I left them open without obvious issues. However, once I had open mesh floors on all my hives, I started to close them off in winter with small pieces of scrap plywood as I believe with OMF's ventilation concerns become a non issue and often bee clusters move up to the feedhole and act as a physical blocking off anyway, so I don't think they particularly like them open.
I don't insert the varroa trays in winter, but get doing it if you prefer to. There seems some observed improvement in early spring build up on solid floors (which can be a good thing or not, depending on your weather that year and the nectar flows you can expect).