He is the one who has the problems, so he is the one who needs to get help. You can refuse to stand by and watch while your boyfriend self-destructs. You can insist, for instance, that he stay sober if he wants to stay with you. Refusing to date someone who has given himself over to drunkenness or other vices is a reasonable boundary for you to have personally. You can’t control whether he gets himself help or not, though. If you don’t know the pitfalls of being sucked in as a co-dependent to someone who chooses to cope with life in a self-destructive way, learn them. That can hardly be stressed too much, both for your sake and for his. If you become a co-dependent, you are no longer helping him with your own feet on solid ground. You would be in the quicksand with him. Heaven forbid that happen!
Demonic possession happens, but the obvious manifestations that we usually think of when we say that is the diagnosis of exclusion. If there are physical things going on, treating those with the normal means the good Lord has given the medical profession is not a sign of a lack of faith in spiritual means. Instead, physical, psychological, and spiritual means of recovering ought to all be used, whether or not the patient believes the malady to be one of those or all three.
Who is giving him the psychiatric medications? Is he doing the work they gave him to do? It is rare for a competent physician to give a patient anti-depressant medications but not give anything for the patient to do to help himself. It is not rare for the patient to fill the prescription and then ignore everything else he or she was supposed to do. This is as true for depression as it is for physical problems such as heart disease.
Encourage your boyfriend to make the changes in his habits that his physician suggested when the medications were given to him. Those changes out to help him in his efforts on the psychological front. Also, keep resisting the temptation to step in and help your boyfriend evade the consequences when he makes poor choices. You ought to feel ready to draw your own boundaries and enforce the consequences for violating them, for that matter. If you enable his bad choices, then you would only be playing into the hand of any evil spirit who has a hand in these issues.
In any event, I would not look at this is in an either/or way. Addiction is the work of the devil. It destroys characters and insulates people from living a virtuous life, destroys long-standing relationships of trust, and generally wreaks havoc with those enslaved by it. Those who become enslaved by it are often trying to self-medicate a different issue that they don’t know how to cope with in a way that works. If your boyfriend is flirting with addictions, if he is feeling cut off from God and from grace, if virtue has lost all savor for him, he has a devil at his heels. Whether it is a direct manifestation or not is almost besides the point.
As ChibiViolet counseled, pray for him. Absolutely, pray for him without tiring, even if things get worse, even if you have to cut off contact, and conversely even if things appear to “get better”. He has been made a target, so call on Heaven to keep him from being picked off from God’s fold.