I'm not allergic to anything or get allergies normally, but I started getting burning and red skin on the outer corners of my eyes. What could it be?

Allergy or infection. Patients with severe allergies get inflammation on the eyelids and around the eyes and at the corners of the mouth. This is a skin inflammation called atopic dermatitis. Try an allergy pill like loratidine and a mositerizer like cetaphil or lubriderm (unscented). If it does not go away or gets worse (more red or weepy) go to your doctor.