Using slender-body hydrodynamics, we study the flow-induced deformation of a high-aspect-ratio elastic filament. For a filament of zero rest curvature rotating in a viscous linear shear flow, our model predicts a bifurcation to shape instabilities due to compression by the flow, in agreement with experimental observations. Further, nonlinear simulations of this shape instability show that in dilute solutions, flexibility of the fibers causes both increased shear thinning as well as significant nonzero first-normal-stress differences. These stress differences are positive for small-to-moderate deformations, but negative for large-amplitude flexing of the fibers.