Outer corner of right eye is red and has been for 3 weeks. It doesn't hurt but sometimes burns and itches. No discharge either, what could it be?

Angular blepharitis. Probably a flare up of some blepahritis. Try warm compresses to the closed eyelids for 5 minutes twice daily, and clean the lid area with diluted baby shampoo. You may need to see your ophthalmologist for treatment with an ointment or drop.