Is it permissible for my brother to mortgage land held by him in trust for shareholders—without their knowledge or permission—for a personal loan, and are they entitled to claim what corresponds to their share of the profits?

Your brother's mortgaging of land he does not own, and without the permission of its owners, is forbidden (haram (definition)). He is liable for it and must release it and return it to its owners. The partners do not have the right to claim a share of the profits from the land he purchased in exchange for the mortgaged land. However, they can demand from him the rent for the land during the period of the mortgage, because his action is considered a type of usurpation (ghasb), and he must return what was usurped and pay a rent equivalent to its مثل (اجرة المثل). If your brother took out an interest-bearing loan (riba), he has committed a forbidden act, one of the major sins, and he must repent to Allah Almighty.