Well, from your description, I have to say that your eyes may suffer from eye infection. Generally speaking, eye infection is the most common cause of burning eyes. So the first thing you can do is to make sure where the eye infection is from. In that way, you can just avoid it. And also, you can take some home remedies to treat your symptom. For example, you can just get a soft cloth with cool milk, and then put the cloth under your eyes. In that way, it will be effective. By the way, having some cool ice or cool cucumbers on your eyes can be tried too.
That the outside corner of your eyes hurts is probably because of the sinus problem. If you have no problem about the sinus, a regular cold may be the cause. Injury can also lead to pain in the outside corner. So you should find out whether you have your eye corner injured, for sometimes you may ignore this fact just because that the injury is not severe. When your eyes are infected, this problem can also occur. The pain in the outside corner can be the symptom or other eye diseases. The main disease that can cause the pain is glaucoma which can finally lead to blindness if left untreated. You can try compresses soaked in hot tea to relieve the pain for the time being. Then consult your doctor.
The feeling of burning, or sore eyes may be caused by a blocked or infected tear duct. As we all know, we have a bit of natural mucus discharge building up in the corners of our eyes every morning. They can be make up of skin cells from our upper eyelids, dust, tears and mucus. There are plenty of bacteria, viruses or fungi in them. We will sweep them out well by our tears. Once the allergens increase or a blocked tear duct occurs, we fail to clean up the eye boogers completely. As a result, we will suffer some eye infections, especially in spring with a unpredictable weather, not always. Never forget our powerful immune system. Well, it is time to clean up your room and keep a good personal hygiene.