Actually a 440V motor can be run successfully on 220V. One of my shapers, an 18 inch Alba, has a 3hp motor which is nominally 460V, and is Delta connected as standard, meaning I can't restrap it for a lower Voltage. One interesting experiment that I have tried is running it from a 230V inverter. It runs fine. Naturally the maximum power is less, but even with only about 3/4 of a horsepower available the machine can remove metal pretty fast. (Running single phase motors on reduced Voltage is not a good idea, they can burn out the start winding if they don't get fully up to speed.)
But that is not the original writers problem. Usually the motor should not run any hotter with the inverter than it would normally. With modern insulation that can be pretty hot. There can be a problem when you run slow for extended periods, especially with a heavy load, since the fan in the motor will be running slow too and will be less effective, as mentioned by two posts above. The heat generated in the motor is pretty much dependent on the current rather than the power, and the current at low speeds and high torque can be high. If you are able to set up a current limit in the inverter, it should be set to the same or a little less than the motor maximum rating.
John