Dry eye: my eyes get very dry and red. I can't keep my eyes open for more than 10 seconds. How can I treat this?

Depends. The main issue in order to determine how to treat this, is to see an ophthalmologist so that they can determine what is actually causing the dryness. Treatments are catered to the cause and not just the symptom.
Treatment. There are multiple possible treatment options for dry eyes. However, you need to determine if you are having problems producing tears, or if you are producing tears which evaporate rapidly off the surface of your eyes, or both. See you eye md for a complete evaluation.
See eye md. Need to find out the cause of this, so get an appointment with your eye doctor. In the mean time try some artificial tear drops, these you can buy over the counter.