At best, if the inverter control circuit has a properly designed and fully working undervoltage lockout function, and there are no faults in and around the inverter output stage, all using a Variac will do is severely stress test its powerup sequence (as mk_ has explained).
To actually bring up an inverter gently, you'd need to isolate the control circuit supply and power them at rated voltage then slowly bring up the DC bus voltage (feeding the output stage). However, odds are, to avoid the control circuit going into a fault shutdown state, there are some monitoring circuits that would need to be tricked to avoid the control circuit detecting the power stage is not operating at the expected power levels, without defeating any overload protection function they may have. Good luck doing that without a full component level service manual + a full set of factory Q.A. test jigs!
All it takes is *ONE* dodgy cap, resistor or catch-diode in a snubber circuit, failing in operation, for the power output stage to destroy itself faster than a dim bulb tester can react. Limiting the current to the output stage with a dim bulb tester helps limit cascading damage: e.g. if an output device shorts, it will prevent it blowing the s--t out of any associated current sense resistor or blowing traces off the PCB. It wont prevent a failed output device going drain - gate short (or collector - gate short if IGBT) and frying its gate drive circuit, though if you are lucky, it will go three way short and the gate drive circuit may survive due to the dim-bulb tester limiting the energy.