The heating of tungsten monoblocks at the ITER divertor vertical targets is calculated using the heat flux predicted by three-dimensional ion orbit modelling. The monoblocks are beveled to a depth of 0.5 mm in the toroidal direction to provide magnetic shadowing of the poloidal leading edges within the range of specified assembly tolerances, but this increases the magnetic field incidence angle resulting in a reduction of toroidal wetted fraction and concentration of the local heat flux to the unshadowed surfaces. This shaping solution successfully protects the leading edges from inter-ELM heat loads, but at the expense of (1) temperatures on the main loaded surface that could exceed the tungsten recrystallization temperature in the nominal partially detached regime, and (2) melting and loss of margin against critical heat flux during transient loss of detachment control. During ELMs, the risk of monoblock edge melting is found to be greater than the risk of full surface melting on the plasma-wetted zone. Full surface and edge melting will be triggered by uncontrolled ELMs in the burning plasma phase of ITER operation if current models of the likely ELM ion impact energies at the divertor targets are correct. During uncontrolled ELMs in pre-nuclear deuterium or helium plasmas at half the nominal plasma current and magnetic field, full surface melting should be avoided, but edge melting is predicted.