Wouter has some good information, but there are more scenarios where not providing a high enough voltage can damage a device.
Some higher end display screens require multiple voltage sources, and failing to power one source to a high enough level, or fast enough, before a second source, can cause damage to the screen or controller.
Some devices with internal mosfet can be damaged by underpowering the source. As was explained by a TI employee about a current controlled led driver, if the VLed source is too low to provide the selected current through a channel, the logic in that channel will try to drive the channel's mosfet harder to try to sink more current. Eventually, the mosfet will burn out, if not other parts of the chip. I wish I could find that discussion and link it.
While not directly causing damage to the device being underpowered, failing to provide the right voltage to a heating element could cause what is being heated to not heat up correctly/fast enough. Winter Water Pipe heaters, electric stoves, microwaves (for a loose meaning of "heater"), certain car parts. Worse, medical devices or heating in artic environments. Same for cooling solutions, like fans or ACs or pelters. A underperforming fan due to voltage issues can cause it's target to overheat. Water pumps as well. And all three can be damaged by the side effects of it. Water pumps normally use the moving water to cool themselves. A lower voltage will cause it to move water, but might not be fast enough to cool itself down. Underperforming fans might be cooked by the device it could not cool down. Heaters themselves might freeze if they cannot get hot enough.
And last I can think of, battery chargers. A malfunctioning charger, or simply badly designed one, as part of a larger circuit, could cause a lower voltage in a charging state. A battery could feed back into the circuit when it shouldn't.