Ok, sorry for the delay.
This film was everything I'd hoped it would be in terms of presenting an alternative stance to the typical, usually 'Hollywood', exorcism film. (Some friends recently made us suffer The Conjuring, it was a fun evening but a dire film.) I've watched quite a few films recently that deal with problems in the Catholic church. Calvary is one of the best films I've seen in a long time (the much more serious follow up to The Guard) and Mea Maxima Culpa is a very interesting documentary.
I have to say I'm pretty surprised that David has concluded 'then you realise she is actually possessed', as it's clear to me the thesis of the film is that there is no such thing as possession and this is a film about why she behaved as she did and why she was responded to in the way she was. If nothing else, the real case on which it was based (as Richard has linked to) is an obvious, horrific, example of someone falsely diagnosed as being possessed, who died as a result, and where the people involved were convicted of homicide.
I agree with what others have said that the film is about (but not only about) the inappropriate reaction to mental illness, in particular by the Catholic church. Assuming you haven't concluded, as David did, that she really was possessed, then it's obvious we're meant to be pretty horrified by the mother, the second priest and the bishop. We're also meant to be angered by the supine and ineffective reactions of the father and the first priest by the end of the film.
As I say, that seems fairly obvious. I think the film on a deeper level is more about how people, especially the vulnerable, can be influenced by others, and also about repression and trying to escape from it. In fairness to David, there is obviously the moment right towards the end of the film when the protagonist has some demonic speech. It seems to me this is not only designed to give a touch of ambiguity but to emphasise the point about how she has been influenced by others.
It's clear early on that she has epilepsy. Her parents are religious and very strict. It seems they are particularly repressive towards Michaela. Her mother tries to emotionally pressure her into not going to university. Any child is 'vulnerable' to religious indoctrination by their parents (read some Dawkins on that); Michaela is especially vulnerable because of her illness. She is still aware that she needs to escape this repression and leave for university but can't break free from her religion, at least initially.
I thought the scene at the disco where she meets her boyfriend and then dances alone, but clearly very free, was especially sweet. It seemed to symbolise her potential to leave the repression from her family. However, she was unable to escape (i) her illness, as she fitted again at university, and (ii) her parents, as she has to return home for the holidays.
Whether through personal weakness or what her parents and religion have done to her, she is unable to handle the pressure of university. The scene where she runs out of an ink reel for her typewriter is particularly distressing, as it symbolises not only her personal struggles but also the breakdown of her relationship with her boyfriend, who is unable to cope with her condition. The irony is that the only coping mechanism her parents have given her - her religion - actually makes her worse not better.
There is therefore something of an inevitability to her descent. She seeks solace in the priests, and again the stronger willed is the more misguided of them, as with her parents. She moves away from seeking a rational explanation for her illness. This is not surprising, given her long term negative experience of doctors and their lack of success and the fear of the serious diagnosis they eventually come up with. Possession offers a simple explanation and one with an 'easy' solution, promised by the Church she has been taught to believe from a young age.
Whilst she initially resists this explanation, she is worn down over time and becomes mentally weaker and yet more vulnerable. She eventually is in a spiral of increasing belief, which exacerbates her 'symptoms', which exacerbates the response from the priest and her mother, which exacerbates her belief, and so on. It seems to me this is the explanation for the demonic voice, as she is unconsciously 'acting up' to the religious diagnosis of possession. As this is not the real problem and the exorcism takes place to the exclusion of her personal care, she eventually dies from the process.