An apparatus comprises at least one processing device configured to generate, for a storage system that is part of an active tier of a storage environment, at least one storage capacity forecast comprising probability values for storage capacity of the storage system over a designated time period. The processing device is also configured to determine whether any of the probability values exceed at least one designated storage capacity threshold and, responsive to determining that a given one of the probability values exceeds the at least one designated storage capacity threshold, to select remedial actions for freeing up space on the active tier of the storage environment. The processing device is further configured to initiate the selected remedial actions for freeing up space on the active tier of the storage environment to prevent the storage system from reaching a designated storage capacity error condition.