Symptoms and risks of driving on bent rims range from increased vibration (especially at higher speeds) to tire damage resulting in a flat tire. If a bent tire rim is suspected, it is important to get the rim inspected by a mechanic as soon as possible.
Tire rims are precisely machined to hold a tire in place and provide a proper seal so tires can remain inflated. Any damage to the rim may throw the center of gravity of the tire off balance, causing it to vibrate. This vibration becomes more noticeable the faster the car is travelling. Even if this is the only symptom, this vibration causes uneven tread wear and put added stress on a car's shock absorbers. If the damage to the rim is great enough, the rim may not be able to hold the air seal on the tire properly or it may physically damage the tire. Both scenarios result in a flat tire, sometimes spontaneously.
As noted by Cars Direct, bent rims do not necessarily require full replacement. A car mechanic can determine if a damaged rim can be repaired or needs replacing. There are methods for straightening rims, but all require specialized equipment and should only be attempted by a mechanic.