The stability of liquid films on moving fibers is studied with a focus on effects caused by film-solid adhesion interactions and hydrodynamic interactions between the film and the surrounding gas. We show that at high fiber velocities (large Reynolds numbers) the film-gas hydrodynamic interactions induce instability in films, which would, otherwise, be stabilized by the adhesion interactions at static conditions. Two types of unstable modes caused by hydrodynamic factors are found: the first corresponds to the Kelvin-Helmholtz waves induced by inertia effects; the second, induced by viscosity effects, is observed at smaller (yet still large) Reynolds numbers when the Kelvin-Helmholtz instability is suppressed. Linear stability analysis of the system of coupled hydrodynamic equations of film and gas flow is performed by the method of normal modes, We derive the general dispersion relation as an implicit equation with respect to the mode frequency. In the limit of vanishing fiber velocities, the obtained equation provides the dispersion equation for motionless fibers. The conditions of film stability and unstable modes are analyzed in terms of two dimensionless parameters, the adhesion factor, which quantifies the intensity of film-fiber interactions, and the Weber number, which quantifies the intensity of film-gas interactions. The Reynolds number determines the regions of validity of the inviscid and viscous regimes of film instability. The results are illustrated by estimates of the stability conditions for moving fibers in terms of the stability diagram, the critical film thickness, the fastest unstable mode, and the corresponding characteristic break-up time, Although the methods developed are applicable for any type of liquid-solid interaction, the estimates were made for the long-range van der Waals interactions. Practical applications include various technologies related to fabrication and chemical modifications of fibers and fiber products, e.g., spin finishing and lubrication.