We investigate how a high current density perturbs the phonons in a biased graphene nanoconstriction coupled to semi-infinite electrodes. The coupling to electrode phonons, electrode electrons under bias, Joule heating, and current-induced forces is evaluated using first principles density functional theory and nonequilibrium Green's function calculations. We observe a strongly nonlinear heating of the phonons with bias and breakdown of the harmonic approximation when the Fermi level is tuned close to a resonance in the electronic structure of the constriction. This behavior is traced back to the presence of negatively damped phonons driven by the current. The effects may limit the stability and capacity of graphene nanoconstrictions to carry high currents.