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.
The bible is not a ‘medical encyclopedia’ so naturally it does not refer to every specific mental problems, but we can assume many came to Jesus to be healed of them and they were called ‘various diseases’. Clearly by observing some people mental problems, in the Bible, even if only temporary, they existed. Why else would Judas have committed suicide if he had not become so fully brought under depression and a sense of doom?
Even in the Old Testament we can see that people generally understood the idea of a person who had mental illness. To avoid trouble for example David pretended to have an extreme mental illness:
David took these words to heart and was very much afraid of Achish king of Gath. So he pretended to be insane in their presence; and while he was in their hands he acted like a madman, making marks on the doors of the gate and letting saliva run down his beard. (NIV 1 Samuel 21:12-13)
Another Old Testament case is where King Nebuchadnezzar actually went insane and than later regained his sanity.
Immediately what had been said about Nebuchadnezzar was fulfilled. He was driven away from people and ate grass like the ox. His body was drenched with the dew of heaven until his hair grew like the feathers of an eagle and his nails like the claws of a bird. At the end of that time, I, Nebuchadnezzar, raised my eyes toward heaven, and my sanity was restored. (NIV Daniel 4:33-34)
As we approach the New Testament there seems to be an upsurge in ‘demonic activity’, possibly as the doctrines for the church were so corrupt that demons had nothing better to do than wage war on the minds and bodies of sinners who had opened a door to them through occult practices or whatnot, we do not really know , nor need to understand the secrets of darkness. In any case Jesus directly cast out many demons as part of his ministry and sometimes mental illness and other diseases were was associated with demonic possession and sometimes not. The Bible therefore clearly distinguishes, between natural mental illness and demonic mental illness.
News about him spread all over Syria, and people brought to him all who were ill with various diseases, those suffering severe pain, the demon-possessed, those having seizures, and the paralyzed, and he healed them. (NIV Mathew 4:24)
After the foundation of the early Church, for whatever reason, possibly just because the church became much more powerful, demonic possession did not maintain the same ‘spike’ of occurrences as it did in the ministry of Christ. However this does not mean that on occasion this is still the cause of mental problems. I have close relationships with people who provide full time pastoral care for Christians and I can assure you many people including Christians suffer mental illness. Many that can be cured, or at least helped by drugs prescribed by doctors, and some, although possibly rare, are also struggling with demons. To distinguish between the two, when scriptures are quoted in relation to the problems a demoniac is suffering, there is a violent reaction to the word of God, but those who are just depressed, or bi-polar, etc. the scriptures help and comfort. Of course typically a worrying depressed by-polar, might worry that they are possessed, even while sincerely seeking God and the scriptures, and the scriptures must then be used to reassure them that such is not the case! If they are believing in Christ and cannot possibly be confessing his name by faith if they were demon possessed.
I have partly tried to offer comfort to people suffering mental health issues in this post, unfortunately to the potential loss of comfort for these dear people; it has recently been recently closed as off topic. Please feel free to re-open as I think that is the more wise and mature approach to responsibly render ones vote, regardless of how ell intentioned those that closed the question may have been. Either way, there is a book I would recommend much better than any post, which honestly deals with the subject, written by a famous English preacher who was also, in a previous career, a respected medical doctor! Well what a surprise someone actually understanding what they are talking about, why not read such a person’s book? It is called Healing and the Scriptures by David Martyn Lloyd-Jones. Anyone seriously interested in the subject, or those who are in the medical or counseling field would do well to study it.
I also speak with some authority on this subject, as I am one of those rare people who did suffer demonic possession and upon believing in Christ was completely and instantly healed, thirty years ago. I also know dear Christian friends who need medicine for mental deficiencies. Some are able to wean themselves off, and others that are not, depending on the condition. We should pray and encourage such people, not judge them, accurse them of being ‘demonized’, or deny their problems. That would be to provide great offence to God! Such people can have very great and impressive faith while struggling with very difficult problems and can be an example to us who enjoy such an easy life in terms of being able to cope in a career and raise our families without having our whole world collapse around us now and again. Thankfully God is able to reach those people and take them out from under the pit, praise his holy name forevermore.
As a former demoniac I certainly had a lot of emotional problems to contend with for many years after becoming a Christian. I have tried to show a way for other’s who suffer similar problems, that are not medical based, here. This post is very dear to me own heart.
I would only add one caution to the subject. The uncontrollable desire to sin which the world suffers and atheistic psychologists and psychiatrists might pretend to manage (compared to their Christian counterparts using biblical principles) should not be diagnosed as a 'mental illness' requiring therapy and or medication. For example, I was a kleptomaniac from the age of six until sixteen, the difference was simply that I was delivered from a life of sin through simple faith in Christ. Most of our generic mental ills are just the ills of sin, which require nothing more than repentance and faith in Christ, followed by a total intent of surrendering our life over to him and his word, both internally and externally. From a general standpoint from our modern perspective we would call a person who enjoys watching others killed as a great mental illness, but in the Bible in the Roman amphitheaters this was just called entertainment and is regular sin. We must limit mental illness to real medically understood anomalies that aredifferent from just intense sinful cravings. Only the church is qualified to administer medicine for sin through the gospel.
The story of King Nebuchadnezzar answers this, in Daniel 4:19-37. A summary of the story: Daniel gives a prophecy (from God) that Nebuchadnezzar:
... will be driven away from people and will live with the wild animals; you will eat grass like the ox and be drenched with the dew of heaven. Seven times will pass by for you until you acknowledge that the Most High is sovereign over all kingdoms on earth and gives them to anyone he wishes.
A key verse is 34a:
At the end of that time, I, Nebuchadnezzar, raised my eyes toward heaven, and my sanity was restored.
Conclusion: mental illness (insanity is implied here) exists in the Bible but can be healed in those who acknowledge God.
So, with mental illness, especially in its most severe forms, there is little ability for the sick person to choose his actions properly. I know this from first-hand experience, when I have fallen into a depressive state (related to bipolar), I am not only less able to think clearly, but I find that I behave in a way which is much different from how I might otherwise behave. I get short tempered, bitter, and sometimes paranoid. In one such episode, I could think of little else but the overwhelming despair.
While depressed, I am more likely to be less than charitable, but that is not something I can choose. I may be responsible for hurting someone's feelings, but that responsibility is necessarily diminished in the same way we would view a child as less culpable.
As to the Bible, well, there is a bit of ambiguity there. The closest I can think of to a parallel is the case of someone being "known" outside of marriage. If a woman is raped (no choice involved), then her culpability is null (Augustine has pointed out that if a woman enjoys it, then there is moral culpability, but that, I think, would be a truly rare phenomenon). While I admit that the situations are not parallel in a strict sense, I cannot help but believe that a similar standard applies.
It's not that the Bible does not have a concept of mental illness (that would be silly, of course there were mentally ill people back in those days) but I believe mental illness is more of a modern term. They usually referred to those who were mentally ill as "the dumb" or more generally, just simply those with "afflictions"
Here are a few examples in the bible of what society today deems as a mental disorder/illness:
