Is it permissible to take out a diminishing interest loan from a bank to pay off partners who wish to exit a successful service project, which is a source of livelihood for a number of people, given the pressing necessity and the inability to secure funds through other means?

It is not permissible to take out a loan for a project from the bank with interest; it is a usurious loan which is forbidden. Interest on all types of loans is forbidden usury, which is not permitted by need or necessity, unless there is an extreme necessity that removes its sinfulness. Necessity is a state of severe danger or hardship, such that one fears harm to life, limb, honor, mind, or property. What has been mentioned has no relation to the necessity that permits committing what is forbidden. If you are unable to purchase the shares of the remaining partners or find an alternative for them, then you must terminate the partnership agreement, and each partner returns with their capital. The loss is proportional to each partner's share, and profits are distributed according to the agreement.