Can allergies cause constant head pressure and tinnitus?

Yes. The accurate evaluation of this problem will require a physical examination by a healthcare provider. Allergy testing can be done by either skin testing or blood testing (rast).
Not tinnitus. Allergic nasal disease (rhinitis) if poorly controlled for several days can lead to obstruction of drainage of sinuses leading to pressure/pain in the head but tinnuitus rarely, if ever, is part of this condition. Advise seeing primary care provider and consider neurology and/or ENT consult.
Absolutely. The ears and ear tubes are intimately associated and connected to the sinuses. And head pressure could actually be sinus pressure.