I have constant pain and pressure around and under my eyes. Is this due to seasonal allergies?

Maybe. The paranasal sinuses are located above, beneath, between, and behind the eyes. Pressure or fluid within the sinuses could cause pain, and this could be caused by allergies but also infections and exposure to irritants. However, constant pain around the eyes could represent another condition and should be evaluated by a doctor.
It could be. Pain, especially pressure around the eyes can be due to sinus inflammation. Try Benadryl (diphenhydramine) or Claritin and see if it helps. It could also be tension type headaches, .