It is possible for you to have burning eyes when you look at the oven because of the radiation. However the possibility is low. Your eyes may get the infection to some degree. You should better have the good rest now. You could also eat more food with vitamin C to moisture the eyes.
Well, that sounds quite weird, because generally speaking, our eyes are fine when the oven is turned on. Anyway, there might be something wrong with your eyes. I suspect it is resulted from a lack of sleep and photophobia, for with that disease, your eyes become sensitive to strong light source and would be uncomfortable. Just try to visit a doctor if necessary.
When the oven is on, there must be heat around it. If you stay too close to the oven, your eyes may be burned by the heat or steam. Stay away from the steam or heat rising from the oven to protect your eyes. Also, check what is in the oven, as you may be allergic to something you are cooking which vaporize when it is heated.