This answer wants to be objective and won't give a plaidoyer pro or contra exorcism.
First some words about the biblical foundation of exorcism.
The idea of an evil power didn't actually play a role in ancient Israel because of the sole and universal power of JHWH – except for two pericopes:
- 1 Sam 16:14-23: Talks of the expulsion of a demonic power that tortures King Saul
- 1 Tob 8:1-9: Talks of a ritual expulsion of demons
The evil doesn't lay in any demonic power but rather in the weakness of man … until in the postexilic period a belief in demonic power develops (but still they don't have any dualistic principle which assumes an equivalent power beside God).
Biblical passages are:
- Ijob 1:6 – 2:7: Describes Satan as a man at the court of God
- Sach 3:1-7: Satan as an adversary and accuser of God
- 1 Chr 21,1: The Hebraic word for Satan evolves to a name which represents God's wrath but is not God himself
- Wis 2:24: Satan as the adversary of Israel
But nevertheless – the appearance of Satan in the OT is a marginal phenomenon.
Today's image of Satan as a spirit being of divine power is later developed in the non-biblical Jewish literature.
Things are different in the NT – e.g. in the Gospel of Mark: Here we find four exorcistic traditions:
They all say: Jesus had the ability to heal from illness and possession. Similar passages can also be found in the other synoptic Gospels (of Matthew and Luke).
From the three summaries in Mark's Gospel 1:34, 1:39 and 3:11 we know that Jesus did perform exorcism and gave this ability the twelve apostles.
Alright, enough Biblical evidence – what about exorcism today? What do say …
Catholic authors (Dr. Holböck, Pater Rodewyk, J.M. Hartmann, Corrado Balducci, a Bishop of Trier (Franz Rudolf Bornewasser), Pope Benedict XVI.) emphasize that there are cases of real possession that cannot be explained by medical doctors and psychologists.
I quote Joseph Ratzinger:
The evil is not only a psychic component, it's rather an oncoming, independent power that accrues humans. [...] Where HE [God] is, appears he [the devil] as void.
Exorcism is still an official doctrine of the Catholic church.
And what do say …
Luther developed – as part of his Christology – a massive idea of the devil.
But nevertheless an idea of exorcism is practically not present1 in the Protestant Church – there's also no ritual like the Catholic "Rituale Romanum".
There are even some who want to erase an idea of an evil demonic power saying it's superstition, like Rudolf Bultmann:2
The church shall hurry up to exterminate it [the idea of demons]
1: With some exceptions, e.g. Christoph Blumhardt
2: Rudolf Bultmann, ''Kerygma und Mythos'' (1948), p.150