If I'm being honest here, I do have schizophrenia, but I've been stable and haven't had an "episode" in over 5 years. But a big part of that is because I have avoided stressful situations as much as possible.
So though it may sound hypocritical, I wouldn't date someone with a serious mental illness, because I don't think it would be good for either of us.
But at the same time, I have been left heartbroken by someone that wasn't willing to look past a stupid diagnosis. So it really is kind of a tough question to answer.