You don't lose your pick, but you could end up at a huge loss.
What happens is, if the time runs out, they can still make their pick at any time they wish. But, the #2 selection can also choose, and once they do, #3 can choose, and so on. What happened a few years ago was once the Vikings passed, teams began to line up at the table and rush picks in, and the whole thing can be a huge mess. If other teams get their picks in before you, you can end up screwed.