Does low power in the mains causes damage to household supplies?
Low voltage can definitively damage electronic equipment, such an induction heating equipment controls.  The low voltage may very likely cause the equipment internal power supply to try to draw more current, which could result in overheating and possible equipment damage. Anything that contains electronics should not be supplied with voltages lower than the rated input voltage stated on the nameplate. 

Resistive equipment such at resistance heaters will not be an issue.

It is not nonsense.