Can dry eyes cause eyelids to droop, like is it a symptom?

Voluntary ptosis. Sometimes due to the discomfort of dry eyes the upper lids are kept lower to reduce friction of blinking. If you are concerned see your ophthalmologist since some serious conditions show up with drooping lids even in a young person.
No. At your age of 20, dropping of the eyelids may be from fatigue more than anything else. Dry eyes will not cause cause your eyelids to droop.