I went to the weather underground website (wunderground.com). The closest places to the Cinque Terre were Pisa and Genoa.
Weather history for the period from February 1-15 (240 days - i.e., the last 16 years):
Sunny - Pisa 26%; GEnoa 21%
Rain - Pisa 21%; Genoa 20%
Partly cloudy was the most frequent condition.
I'd guess that most of the shops and restaurants close up then, since my experience in looking for hotels in February in other parts of Italy in the last 4 years is that many of the smaller ones and B&B's close for February either for renovations or for the owners' vacations. Or maybe because it's low season, it just isn't economical to keep them open. If it's overcast the hiking won't be scenic (the main reason to visit) and if it's raining, parts may not even be hikable. If you can spontaneously spend a couple of days if the weather's good, go for it. If you aren't flexible enough, best to save it for a better time of year. (in my opinion)