Electromagnetic interactions between colliding heavy ions at the Large Hadron Collider (LHC) at CERN will give rise to localized beam losses that may quench superconducting magnets, apart from contributing significantly to the luminosity decay. To quantify their impact on the operation of the collider, we have used a three-step simulation approach, which consists of optical tracking, a Monte Carlo shower simulation, and a thermal network model of the heat flow inside a magnet. We present simulation results for the case of ion operation in the LHC, with focus on the ALICE interaction region, and show that the expected heat load during nominal operation is 40% above the quench level. This limits the maximum achievable luminosity. Furthermore, we discuss methods of monitoring the losses and possible ways to alleviate their effect.