It depends on the
AC drive. Some drives can be set to indicate a loss of input phase as a fault condition, even if a run command has not been given, and some drives only display fault message in heavy load. Suffice it to say that a traditionally sized drive will fault rather quickly upon loss of input phase, with a run command, since the DC bus voltage would most likely drop below an acceptable level. I have seen an application where an AC drive was deliberately oversized, so that it could run the motor at full load with only two phases input. But this was a very special application and is not generally practical.
In my experience, lack phase running is not good for the drives, the temperature will rise to harm it's service life.