The following is a brief synopsis from Jay Adams regarding how we may be sure, Biblically, that a believer cannot be possessed (or "oppressed") by a demon. Note: while I do not completely agree with Dr. Adams' amillennial views, including the teaching that Satan is currently bound and therefore demonic activity during the Church era is non-existent, he is certainly correct in that addictions and life-dominating sins are not due to such possession. I have counseled many women who actually believe they are inhabited by "demonic forces" compelling them to starve/purge/cut, but this is nothing more than an esoteric dodge of personal responsibility.
Yes, Satan is ultimately at the root of all sin; and eating disorders are certainly sins of both thought and behavior. However, demon possession in the Gospels was a malady - Christ cast them out as a ministry of mercy. Where sin was involved, He called men and women to repent. His answer to habitual or besetting sin is the same now as it was during His earthly ministry. There is absolutely no Scriptural warrant for attributing addictions or strongholds of sin to demonic forces.
"There is no Biblical reason to think that demonic possession (or oppression) can occur in the life of a Christian. The simultaneous presence of the Holy Spirit, Who dwells within every true child of God, and an "unclean spirit" is impossible. This is clear from the utter antithesis of the two noted in Mark 3:20-30.....I have seen incompetence in counseling excused by resorting to the diagnosis of possession by demons, sometimes with very damaging effects. If, for example, one's problems are the result of his own sinful behavior, and they are instead charged to possession by an evil spirit, those problems may be complicated rather than solved by efforts to cast out a demon. Not only will such efforts fail, leading often to hopelessness and despair, but they will shift the focus from the counselee's own responsibility. He will be viewed as a helpless victim rather than a guilty sinner. The results are likely only to confirm him in his sinful life patterns, and the frustrations of counselors who are reduced to fruitless prayer and pity are likely to encourage deeper depression and even despair." (Jay Adams, "The Big Umbrella", currently out of print)
While I do believe demons still hold an active influence over this world, tempt believers to sin and believe lies, and can inhabit non-Christians, they cannot compel anyone to start or stay stuck in an addiction (despite what Nancy Alcorn claims). If we follow the Biblical model, we have to take responsibility for our own behavior before God - and stop giving the enemy so much credit.
Also see my posts on true spiritual warfare, and why "deliverance ministries" are not biblical: