We perform a local (short-wavelength) linear stability analysis of an axisymmetric column of magnetized plasma with a nearly toroidal magnetic field and a smooth poloidal velocity shear by perturbing the equations of relativistic magnetohydrodynamics. We identify two types of unstable modes, which we call ‘exponential’ and ‘overstable’, respectively. The exponential modes are present in the static equilibria and their growth rates decrease with increasing velocity shear. The overstable modes are driven by the effects of velocity shear and dominate the exponential modes for sufficiently high shear values. We argue that these local instabilities can provide an important energy dissipation mechanism in astrophysical relativistic jets. Strong comoving velocity shear arises naturally in the magnetic acceleration mechanism, therefore it may play a crucial role in converting Poynting-flux-dominated jets into matter-dominated jets, regulating the global acceleration and collimation processes, and producing the observed emission of blazars and gamma-ray bursts.