What is an overview of how Christians discern genuine cases of demonic influence from mundane mental health problems?

I think there is a way to answer this question without delving into the speculative, or into individual counseling but we must be careful. It basically amounts to how a person responds to scripture. There can be no consensus on the topic in detail because even when regarding physical diagnosis doctors, expert doctors, may have many different conclusions. When we add psychological and demonic, there will be even less consensus.

What can be said has been outlined by some although it seems a little bit rare because few have broad knowledge to comment. One example of differentiating the types of illnesses and how to treat them was done by a man called Dr. Martyn Lloyd-Jones who was both a reputable medical doctor in England and also became a renown theologian. He wrote a book called ‘Healing and the Scriptures - Including a look at: The role of medicine and doctors, How to combat the demonic, Identifying psychosomatic illnesses, Advice for prayer counselors.’ It’s a little outdated now but contain the essential answers to the question at a more objective and opinion that probably has a least the grains of a consensus among most Christians that believe in demons while also believe in modern medicine.

He dives the grouping of people under these categories and provides examples to discern the typical features that might indicate the proper grouping. I am intentionally in answering typically only because as I said there is no consensus in eth details as that depends on the situation, the wisdom of the caregiver, and many other variables that would delve us into speculative topics outside the purpose of this website.

Physical Illness This seems more obvious a category but people may still confuse it with spiritual:

“Another example is the great Charles Haddon Spurgeon. He used to have periodic fits of depression, when he felt he was not called to preach and sometimes that he was not even fit to preach. What was the matter with him? Well, he suffered from gout,” (P149)

Spiritual Illness Basically he means those problems that can fully be cured simply by believing scripture:

“It will be clear to you that the important point is a diagnostic one. I have always found that with persons in this spiritual category there is a clear diagnostic point. They always show a readiness to listen and they almost jump at any of the verses quoted which give them relief. They hold on to what will really bring comfort and release.” (P152)

Psychological illness

Under this category he summarizes a good point that again focuses on their response to scripture:

I think that you will find almost invariably that those who are mentally ill do not really listen to you. You quote Scripture, they do not listen. They keep repeating the same statements and give the impression that they are waiting for you to finish so that they can say their piece over again. This is almost invariable. You notice the difference as compared with those in spiritual trouble. The latter are anxious to have help. The others are not. I always feel with them that I am a kind of tangent to a circle. One never penetrates, they are almost impatient and go on repeating the same thing. (P158)

Demonic Oppression

The Dr. uses this category to identify the attacks the Devil has made in the past on holy mean and woman of God that may make them depressed or unusually disturbed but have nothing to do with possession that are related to ones own occultism or welcoming of spiritualism into your own life.

Demonic Possession

“What are the diagnostic points in these cases? You generally find a history of dabbling with spiritualism or the occult in some form. It may have been back in their childhood, or during teenage, that they have been introduced to the occult and experimented with occult phenomena. They may also have experimented with drugs.

One clear diagnostic point is that one becomes aware of a dual personality. There is another person. You see it in the face, hear it in the voice. It is an unnatural and quite different voice and can very often be accompanied by horrifying facial expressions. There is also-a most important point-an alteration between what we may call a normal and an abnormal element. These persons can be one moment quite normal and can discuss things quite readily for a time; then suddenly they change. The "other" person seems to take charge. They will tell you that they are conscious of suggestions and voices; and frequently that they have come.to have abnormal powers. In my experience there was a woman who was able to hold a complete conversation with a man in Swedish, a language of which she had never learned a word.

A still more significant pointer is their reaction to the name of our Lord. I always tell ministers who are confronted by the duty of treating such cases to use the phrase “Jesus Christ is come in the flesh" and to note the reaction. Talk to them of "the blood of Christ" and you will generally find that they will react quite violently to this. (P 165)

It should be noted that I am quoting just one man’s opinion, although a credible man, it is not meant to be a medical guide exactly – yet does illustrate the point of consensus:

There are problems in human lives stemming from physical, psychological and demonic. The key differentiator is how they respond to the scripture. Sometimes you do not know what the problem is until we see what heals the problem.