Some of the lower-end inverters put out a modified sinewave, but many seem to have an output almost approaching a perfect squarewave!
I'd guess it could well be responsible for damaging the charger, especially as on modern tools the charger is likely to be a switched-mode type. That squarewave from the inverter will be very rich in harmonics, and could well cause problems for a SMPS.
Now, if you could just get the inverter to run a charger that's connected to the battery that's running the inverter.....