We study the thermodynamics of a quantum measurement-powered engine that converts energy injected by measurement backaction into work. We consider an engine with a finite-dimensional working substance driven by quantum measurements, i.e., by bare quantum measurements, without feedback control or thermal contact in the thermodynamic cycle. On the basis of a Poincar'e-like recurrence theorem for general quantum channels, we prove a no-go result for work extraction from such an engine in the steady regime. In the steady regime, quantum measurements become nondisturbing and do not inject energy into the working substance. Consequently, no work can be extracted. This result reveals the necessity of an entropy-decreasing process, such as feedback control or thermal contact, for work extraction in steady-cycle measurement-powered engines.