What happens if a 240V appliance is connected in a 120V AC power supply?
Probably nothing serious would happen as has been mentioned already but there is a possibility of a potential serious situation. Consider an appliance only intended for operation on 240V AC but is able to work from (say) 200V to 250V. To do so might mean it uses a switch-mode power supply to regulate the internal DC voltages. Let's say it required 100 watts internally, maybe some form of audio amplifier.

At 250 volts AC it would draw 0.4 amps plus 10 % more for inefficiencies - that's a current of 440 mA. At 200 volts AC it would draw 550mA. At 100 V ac it would try and draw a current of nearly an amp if it were able.

The point is that it will try and draw more current at a lower AC voltage and this could blow an internal fuse or damage the switching transistor - the average current may only be 1 amp but the switching current might be 10 amps. Also, at a lower voltage (with the increase in current) the reservoir capacitor after the bridge rectifier will be struggling to maintain low ripple and between cycles the dc voltage before the switching element may sag to only 50 volts - this means a higher instantaneous current draw on a cyclic basis and possibly more damage to the regulating switching transistor.