The more prevalent problems happen at 380V and up. VFDs can cause high voltage reflected wave spikes that are theoretically capable of being 2.5 to 3x the line voltage. So when an old motor was built with 1000V insulation and used on a at 480V VFD, the VFD spikes could be over 1400V, so its only a matter of time before the insulation fails prematurely. But at 240V they still used the same 1000V insulation, so although the reflected wave spikes are possibly 3x the line voltage, that's still below the insulation level of the magnet wire.
There are other possible failure modes though, such as the PWM pulses creating a voltage across the stator and rotor that flows through, and damages, the bearings and races. "Inverter Duty" motors often take measures in the design to prevent this, but you can also find retrofit solutions as well. If the motor is custom or difficult and expensive to replace, I would suggest adding a dV/dt filter to the output of the VFD, not just a reactor (a dV/dt filter is a reactor + additional RC filtering for the spikes and capacitive coupling issues that can cause the bearing damage). You can also get shaft grounding devices to add to the motor that give that voltage a safer path to ground.
A common way to approach this though, especially if it's easy to replace, is to use your existing motor until it fails then replace it with one that is rated for inverter use.