Is it possible to relieve dry, burning eyes?!

It is! Actually there are several different types of dry eyes each having a different cause and methods to treat its symptoms. My advice is to go visit your optometrist and let them know your concerns and symptoms. They will perform a health assessment of your eye and determine the type and cause of your dry eye. If you are just looking for some drops to use throughout the day, I would recommend some artificial tears (Systane Balance, Blink, or Optive are some that I often recommend to patients). All of those are available over the counter.

I would generally avoid drops like Visine or Cleareyes that "get the red out". Those are more vasoconstrictors which simply constrict the blood vessels in your eyes for a short period of time. The problem is, you can get a rebound effect and your eyes can become even more red than they started.

If you were just getting something to soothe some dryness throughout the day, your best bet would be those I mentioned.

I'm an optometry student

Pick up some Visene at the drugstore/grocery store! It relieves drying/burning as well as redness. They really help to freshen up my eyes and make them feel better!