Well, first, you should know that for your situation, it can be because of the lack of sleep. Generally speaking, when you are lack of sleep, your eye will suffer eye strain. And then it can lead to some eye symptom, such as the dry eyes, burning eyes. And in some cases, red eyes can occur too. On the other hand, it can be caused because your eyes are too dry, for when your eyes are dry during the sleep, you will be possible to have burning eyes. At this moment, you can just take some eye drops to your eyes, so that it can moisten your eyes. And then the cool ice can be effective too. Anyway, you need not to worry about it, and it can be fine.
Since your eyes are itchy and red, you probably have eye infections of inflammation. The possible inflammations are Blepharitis, conjunctivitis, corneal ulcers (ulcers on the outer cover of the eyes) and uveitis. If you have Blepharitis, your eyelash will swell along the eyelid. Conjunctivitis is usually known as "pink eye" and it is an infection of the tissue connecting the eyelids and conjunctiva. Corneal ulcer is caused by bacterial or viral infection on the outer cover of the eye. Uveitis causes a swelling uvea due to autoimmune disorder, infection or toxins. When you have this uncomfortable feeling, you should consult a doctor as soon as possible to get timely treatment.