Does Paul ever teach that demons are active in the life of a believer?
Paul does not teach that demons are active in a believer's life. He never once uses the word of demons as being active in the life of a believer; instead, he speaks of the power of His resurrection. The believer is directed to focus on resurrection power, causing demonic stuff to flee.
Les explains this in response to a lady who was all shook up, saying she saw demons in her home and family and was up against the forces of Satan. Les told her her mind was in the wrong places. He urged her to put her mind on the power of the resurrection, and all the demonic stuff would flee like chaff in a windstorm. He said to forget about the demons. He further stated that Paul never once, as far as he knew, even uses the word of demons; if he does, he wants to be told. But he does not believe Paul even once uses the word of demons as being active in the life of a believer. Instead, all Paul talks about is the power of His resurrection.