I have a ceiling fan on our bedroom's vaulted ceiling. Since we don't want 7' pull-cords, the speed is stuck at full-tilt, which is no good. The fan has separate switches on the wall for lights and fan on/off, though, so I tried putting a dimmer switch on the fan. Can that do any damage to the fan's motor, or is there anything else I'm not thinking of that could be a problem? I've noticed the fan makes a little bit of a groaning noise when I have the rheostat down, but since I've never heard the fan at slow speeds, maybe that's normal.
Check the fine print
To find out if the dimmer can work with the fan, you'll have to inspect the dimmer. For this, you'll have to remove the cover plate and possibly pull the switch out of the box (in which case, make sure you shut off the power at the breaker). If you see the text "For Incandescent Only"; or something similar, you should not use this dimmer with a motor load.
Using a standard dimmer with a ceiling fan can damage the motor, and/or cause the dimmer and/or motor to overheat and cause a fire.
There are a few better options available to you.
Install a fan control switch
There are many different fan control switches available; with a wide range of styles and functions, you should have no problem finding just what you want. A simple control like this 3 speed fan control, will run you about 20 bucks, and installs using the existing wiring.
Install a remote control
If you're up for a bit more wiring, there are universal remote control kits available. You'll have to wire the receiver into the fan housing, so it's a bit more complicated than installing the wall control. But once the receiver is installed, you can place the wireless remote anywhere you like. There are a ton of these things available, so you shouldn't have any trouble finding one you like. You can even get a cool touch screen control, if that's what suits your fancy.
Install a new fan
If installing a new fan is an option, you could simply select one that comes with a remote control.
Under powering can certainly damage electronics, including motors.
It's possible, though, that your fan might be fine.
I would look for a fan designed to be used with a dimmer, find out what kind of dimmer it needs. Alternatively, many fans have their own integrated speed controllers.
I don't know but I suspect a dimmer is a bad idea, but you can get 3 speed fan controls that are designed for this purpose. For instance this one.
Common AC electric motors have a high impedance on starting because velocity is 0, This draws a large initial current to get the motion started and as velocity increases impedance drops and the current draw decreases.
A dimmer will reduce the wattage to the fan motor meaning that you are increasing the impedance over time, meaning that heat will buildup until it becomes a potential fire hazard. This is the same reason that you shouldn't use a normal dimmer switch for incandescent bulbs as they have a large initial current draw.
In other words, don't do it. They make smart switches that can sometimes do this in a safer way.