Yes, it is normal for you to get burning eyes because of bleach fumes because of the irritating stimulation. You'd better wash the eyes with clean water to make them look comfortable. You could also go to use some eye drops to release the pain of the eyes. The warm compress could also be beneficial. You could try the above ways.
Well, yes, bleach fumes can lead to burning eyes. And generally speaking, it is normal to have this kind of situation, for your eyes are very sensitive to bleach fumes. And in most cases, they will water excessively when exposed which gets rid of most of the bleach. Of course, burning eyes can be another symptom, so you need to take some measures to treat it. For example, you can just use over the counter lubricating eye drops to help flush out the rest. And also, trying some ices can be effective too. If there is any problem, just go and see your ophthalmologist.
Well, so sorry to hear that your eyes are damaged by those bleach fumes, because as you can see, those products are made up of chemicals, which could be of great danger and irritation to your eyes, leading to some serious problems and consequences. Thus, I suggest that you take some immediate medical treatment in the hospital, and never rub your eyes, before it is too late.