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.