Well, yes you could, but I don't know why you would want to. When I used drains (not needed any more due to quilting sutures), I always numbed up the drain site. Otherwise it can hurt quite a bit having the drain pulled. Patients who knew someone or had seen someone else have that done without anesthetic did not like it at all. Ouch! Also, drains leave an opening that heals much better if it is closed. Until it seals, more drainage can come out on your clothing. So I would put a few sutures or a staple to make that opening heal as a narrow scar. That requires anesthetic for a comfortable experience. Also, the decision to take out a drain, and when to take it out, is a clinical decision that your plastic surgeon makes. Sometimes, when there are two drains, only one will be removed and the other kept a while longer to ensure drainage and decrease the risk of seroma. So for all these reasons, it is best if drains are removed at your doctor's office.
I would not recommend this, because a physical exam prior to removal ensures that an additional problem does not exist. Sometimes, a person can have fluid and low drain output because a drain is clogged. That is just one of many examples. Please get this removed at the doctor's office.