Can untreated nasal allergies make you feel constantly lethargic?

Yes: Yes. In people with chronic allergies involving the nose and upper respiratory tract, lethargy is a common complaint. This may be due to sleep disturbance associated with nasal obstruction or just a part of the allergic process. Treating the allergies with non-sedating antihistamines and/or prescription nasal sprays can be very helpful in improving all the symptoms including the lethargy.

Sure: Allergies are a derangement of the immune system which can make people feel very lethargic especially if symptoms are not well controlled.