It doesn’t hurt to leave it up in the winter. It can provide a place for the BB’s to roost during the winter evenings.
Just watch it to keep it clear of nesting material. I have noticed that my nestbox is not occupied every night during the winter months. The birds will use it when their “weather instinct” tells them that they need additional shelter.
BTW – those young bluebirds on the wire could have been yours.
David
Stafford, VA