What does the Bible say about people with mental illness? Some people with mental illness might do lots of things that are unpredictable and considered sin, up to killing themselves, or possibly forget anything about Jesus and salvation.

So, what does the Bible say about them as in will they still be saved?
The Bible does not really have a concept of mental illness in the sense we have it today. There are, as far as I know, no stories in the Bible that clearly talk about the mentally ill.

There are of course cases of people described as "demon possessed" or "having an evil spirit". Many have speculated that these are people who would today be described as having a mental illness. Examples that spring to mind include the demon called "Legion", (Mark 5 1-20), Luke 4:31-37, The Philippian slave girl (Acts 16). 

Whether we consider this a real demon or a mental illness, one thing that all these stories have in common is that after the demon has left (or they have been cured) no blame is attached to the person. We don't read that the people have to repent, or are forgiven. We could reasonably infer that they are not responsible for what they have done.

ADDITION:While not strictly within the scope of the question, I found while researching this article which discusses the moral responsibility of the insane from the Catholic viewpoint. Its conclusion is that there are kinds of insanity that remove the culpability of those that commit sin, chiefly those of a delusional nature or which prevent the sufferer knowing right from wrong. However if the sufferer still knows right from wrong they may still be responsible for their sins.