Possibility of a brain tumor? My head is constantly in pain and the headaches are getting increasingly frequent...

unlikely. A tumor is an unlikely cause of headaches but constant headaches need evaluation and proper treatment. See a neurologist if possible.
Possible, not the #1. Brain tumors can certainly cause headaches, and increasing frequency of headaches is worrisome and is a reason to see your doctor. Brain tumors are uncommon in 16 year olds, so this is hardly the most likely diagnosis. You should ask your doctor for a workup including a brain MRI and to find the real reason for your headache.