Well, of course, you should know that there are many factors which can lead to burning eyes. For example, according to some experts, eye infection, allergies, irritation can just be the most common reasons, and that is also why your eyes will feel burning when you close your eyes. By the way, high eye pressure can be another factor too. To treat it, cold compresses can be a good way. For example, you can put some cool ice on your eyes, so that it can relieve your symptom. Also, cool tea bags and cucumber can be effective too. Besides, nutrients intake is also important. So you can just eat a diet rich in fresh fruits and vegetables, which provide lots of antioxidants, which may help boost your immune system, especially vitamins A and C and then make your symptom. Anyway, just have a try.