Active search is a setting in adaptive experimental design where we aim to uncover members of rare, valuable class(es) subject to a budget constraint. An important consideration in this problem is diversity among the discovered targets -- in many applications, diverse discoveries offer more insight and may be preferable in downstream tasks. However, most existing active search policies either assume that all targets belong to a common positive class or encourage diversity via simple heuristics. We present a novel formulation of active search with multiple target classes, characterized by a utility function chosen from a flexible family whose members induce preferences for label diversity among discoveries via a diminishing returns mechanism. We then study this problem under the Bayesian lens and prove a hardness result for approximating the optimal policy for arbitrary positive, increasing, and concave utility functions. Finally, we propose an efficient, nonmyopic approximation to the optimal policy for this class of utilities and demonstrate its superior empirical performance across a wide variety of experimental settings, including drug discovery.