A maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables. Such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal-agent problems. By virtue of a classical spike variational approach and a filtering technique, the maximum principle is obtained, and the related adjoint processes are characterized by the solutions of forward-backward stochastic differential equations in finite-dimensional spaces. Then our theoretical results are applied to study a partially observed linear-quadratic recursive optimal control problem. In addition, for the case with initial and terminal state constraints, the corresponding maximum principle is also obtained by using Ekeland's variational principle.