Nobody can answer the "is it too [anything]" questions but you.
Most people would say it would be miserable and crazy to expect to sit outside at a cafe in late December (I would probably agree) but if someone is determined to do that, you can sit anywhere you want (you might have to find your own chair and table since most were probably put away and locked up after the tourists all disappeared in the fall).
Note that no towns literally shut down, ie put up chains across the streets with signs saying "keep out". Many (maybe most?) businesses that cater primarily to tourists might actually shutter for the off season, but some stay open mostly for locals and the occasional lunatic tourist, so there's nothing (other than common sense) to stop you from trying to dine al fresco along the Adriatic on New Years Eve.
It all depends what you want/need/are willing to settle for. Personally, I wouldn't go to these places at that time of year - especially if I wanted to sit outside. I'd either go at another time (even off season, not just that far off) or I'd go someplace else.