Two questions get collapsed into one and shouldn't be. First: did exorcism end with the apostles, or does it continue as ordinary pastoral ministry? Second: can a born-again believer be troubled by a demon at all? This site's tradition answers the first from Mark 16 and the second from what Scripture says about the sealed spirit and the ground a believer still holds back.
On the first question, this site's tradition holds that Mark 16:17-18's "these signs shall accompany them that believe... in my name shall they cast out demons" describes normal Christian ministry today. Derek Prince's They Shall Expel Demons (1998) is the standard modern handbook: confession, repentance, renunciation, and command in Christ's name, followed by ordinary discipleship: deliverance as a normal pastoral tool, not a spectacle.
On the second question (can a believer be demonized), the position taught across much of the deliverance-ministry tradition is precise, not sweeping: a believer's regenerate spirit is sealed and cannot be indwelt by an unclean spirit, but unsurrendered ground in the soul or body (old wounds, ongoing sin, generational patterns) can still give a demon a foothold there, requiring deliverance alongside repentance.
That precision comes from the texts themselves. The believer is "sealed with the Holy Spirit of promise" (Eph 1:13-14) and his body is the Spirit's temple (1 Cor 6:19), so the regenerate spirit is God's own ground and stays His. Paul still prays for the whole man, "your spirit and soul and body," to be sanctified and preserved entire (1 Thess 5:23), and he warns that anger carried past sundown gives "place to the devil" (Eph 4:26-27). The center is sealed. The soul and body are still being sanctified. Ground given to the enemy is taken back by repentance and by the name of Christ, which is exactly what deliverance ministry does.
James's warning stands over the whole question: "the demons also believe, and shudder" (Jas 2:19). Belief is not the issue; submission is. "Be subject therefore unto God; but resist the devil, and he will flee from you" (Jas 4:7) is the pattern this site's tradition commends: ordinary, unspectacular, and available to every believer, not a specialist's trade.