- Any shallow cut, whether made by paper or a sharp plastic edge, hurts more than a deep cut because the shallow one remains open. The body quickly seals a deeper cut with a blood clot. An open wound, no matter how small, hurts.
- Due to all the stuff in paper, some of it is left behind in the wound. The foreign matter (paper is made of wood mulch and a variety of chemicals) stimulates pain receptors in your finger, causing the stinging sensation and causing the cut to heal slowly.
- There may also be a psychological component. We don't expect paper to hurt us, and we get angry and upset when it does. Paper cuts come unexpectedly, when we are engaged in what we think is a low-risk activity, and we can get impatient with them when they heal slowly.
- (Added by Lorna) The paper wants to hurt us.
I think #4 makes the most sense.