I don't think a person's MBTI type would change, although it's important to keep in mind that not everyone is 100% certain of their type. Many believe they are a certain type, but may really be another type and not know themselves well enough to see that error. Since MBTI deals with preferences, behavior can change without type changing since it's likely that one's behavior would not be in line with their preference when under the influence of a cult.
So, a person may prefer to rebel against the cult's psychological manipulation, but may be too weak to do so. For example, an INTP may prefer to question the cult's "authority" and refuse to follow no matter what (in typical INTP fashion), but may instead go against their preference and follow because they have weak self-esteem or because the cult has threatened them and they feel afraid to act according to their preference.
Naturally, I think people are best able to "be themselves" in situations where they are happy, comfortable, and feel supported by others. A cult is unlikely to genuninely produce such a sense of well-being. So, in that sense, you could say an INTP wouldn't really "be" an INTP in terms of behavior. If they feel stressed, they may "act" ESFJ or ENTJ. They may also try to chameleon the cult's leader in the hopes of loosening his/her control by proving to themselves and others how easy it is to seem so charismatic, thus exposing the leader as nothing special.