How long can a lumbar puncture/ spinal tap headache last?

Days. Varies, but often can take a few days to completely go away. If the headaches are increasing or accompanied by other symptoms (stiffness, fever, neck pain, etc) you should see the physician that did the spinal sampling. It takes a bit of time to restore the volume of fluid drawn during that time headache (alone) are not uncommon.
Few days. Usually stop after few days, lay down that will help if the headache got bad go back to see your doctor.