During an average winter, what part (if any) of the Mississippi River becomes unnavigable?
Upper Mississippi river near St Paul Minnesota is closed December - March.  US Army Corp of Engineers is responsible river navigation in the United States.  From winonapost.com, (from 2015) the article states that all vessels need to be south of Lock and Dam 9, near Lynxville, Wisconsin.  Sounds like the upper river is closed so locks can be dewatered and maintanence performed.  

My guess is most of the Mississippi river remains open to navigation south of Wisconsin during the winter. 

US Army Corp of Engineers river navigation site.