There could be damage. With half the RMS AC voltage, that is, half the force pushing charge through the device, we might expect half the current flow. If the device acts like a simple resistor, that's exactly true. That means 1/4 of the normal amount of power is used by the device.
If the device has capacitive or inductive reactance, and has nonlinear effects, then no. Still, with no specific device as subject discussion, we may as well assume one quarter the power usage.
If that power is primarily working a motor, then the motor will be spinning slower. (duh.) Some motors depend on spinning at a high speed to keep themselves cool. If it's not spinning fast enough, maybe it won't keep itself as cool. But at (probably) 1/4 the power, it's not getting as hot, either. Will friction or load keep the motor from spinning at all?
Whether the cooling effect is diminished in the same proportion as the motor heating, depends on the actual type of appliance, the load the motor is pushing, the presence of voltage regulating circuits, and for all I know, the appliance's astrological birth chart.
That's just considering basic motor physics. The range of parts and physical phenomena in a generic unspecified household appliance is vast, and so it is not possible to rule out some other way that half-voltage input could cause damage.
Short answer: without further info, it's guesswork, but the range of guesses must include the possibility of damage.
There is only one way to find out, assuming you can make the plug fit the socket...