I agree - I wouldn't call it obsolete or even archaic, but I wouldn't use it seriously. If, for instance, I wanted to have a serious talk with my daughter who had been moping around the house for two weeks because her boyfriend had dumped her, I wouldn't say, "Now honey, I know you feel awful. You're just got to stop pining for John. I know it's hard, but you have to try."
The movie's title is "Cold Case", so it's not that old as you'd imagine, but somehow or other they used a word that is way past its sell-by date, considering that it was in use in the early nineties, and it comes as a complete shock to me ))))
So, "pine" is out then ! (out of my vocablurary, meaning that I will never use it again).