The dynamics of collapsed polymers in shear flow is very different from that of sheared polymers in good or Theta-solvent solvents, because cohesive attraction between monomers opposes hydrodynamic drag forces. Using hydrodynamic simulations and scaling arguments, we show that a polymeric globule displays a well-defined stretching transition at a threshold shear rate (gamma) over dot*. Below this critical shear rate, the chain remains in a slightly deformed but compact state, while for shear rates larger than (gamma) over dot* the globule undergoes periodic unfolding-refolding cycles. The mechanism for this shear-induced instability is associated with the presence of thermally excited polymeric protrusions and thus very different from the classical scenario of hydrodynamic instabilities at the interface between two viscous liquids. In fact, it turns out that the critical shear rate depends sensitively on temperature and on the microscopic cutoff length scale, i.e., the monomer size. Our results demonstrate that proper inclusion of hydrodynamic effects is crucial: in the free draining case we find (gamma) over dot* similar to R-1, where R is the globule radius, while in the experimentally relevant case including hydrodynamic interactions (gamma) over dot* similar to R. For partial drainage, a crossover between these limits is obtained. The power spectrum of the stretching dynamics for collapsed polymers is very different between the nondraining and free-draining scenarios, in contrast to noncollapsed polymers. The shear viscosity of a dilute solution of collapsed polymers displays a clear signature of the unfolding transition at the critical shear rate and exhibits shear thinning for all shear rates considered. We also propose simple scaling arguments for the shear-induced instability of a planar interface between a polymer melt and a viscous fluid.