How do doctors remove lost sutures?

I'm not quite certain what you mean. If a skin suture, which should be removed but either was missed or left too long and has gotten overgrown, needs to be removed it is unfortunately largely a question of digging around for it. Gently. And preferably with local anesthetic on board.

If you mean how do they remove a needle that got dropped into a body cavity - they look around, carefully, sometimes using magnification (special glasses or an operating microscope.) If they are not certain that a needle was lost in a patient, they may do an Xray (ideally before finishing the surgery and closing up.) Sometimes very tiny needles get lost and left behind and you have to balance the relative risk of leaving it there (generally very low) against the risk of going back in to remove it (again, low, but exposes the patient to another anesthetic, infection risk, bleeding etc.)

Many sutures left in the body are dissolvable and do not need to be removed. If you mean a suture that has migrated away from a dental repair, or something like a hysterectomy, it probably does not need to be removed but will just go away on its own.