The primary answer to this should always be "ask your teacher". That having been said...
There are several reasons you may have redness in your eyes. The first thing that I feel is a sensitivity to chemicals or environmental conditions. I personally stay away from shopping malls, cosmetic counters, and the candle sections as they are loaded with things I simply do not want to breath. Cigarette smoke could also be an irritant. Or it could be dust. Try a hepa air filter at home and see how that goes. Also stay hydrated and rest/eat properly.
As for Shirsasana, an answer would depend on how you are doing it, whether you have the actions in the body for doing it, and for what duration you are doing it. Without that information I could not determine the integrity of your pose.
Generally speaking, Shirsasana is avoided by those who have glaucoma, brain tumors, high blood pressure, or recent cataract surgery. There are other inversions, of course. I would advise asking your teacher this question and having she/he modify you accordingly. It is the teacher's job to balance efficacy and safety for the student.
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