They depend on the cause, sight and size of the problem. Though not all headaches are because of tumors but all most all tumors cause headache depending on their location and nature. Migraine headaches are usually located to one or the either side, common in females in younger age group and cluster headaches are more so in males.
Conditions such as Benign Intracranial Headaches (BIH) are more dangerous as if not picked up and managed on time can lead to complete blindness. Similarly, glaucoma though is an eye related problem but it can present with severe headache as primary manifestation and can lead to significant visual impairment if not treated on time. Sinusitis again can present as dull aching headache with other associated symptoms such as parosmia, nasal stuffiness and tender sinuses but rarely is a cause of headache alone.
Picking up the right cases can of course save lots of irreversible damages ...