Yes, but probably not for very long. If the alternator warning light is on, it means your charging system is not producing enough current to meet the electrical needs of your vehicle and to keep the battery charged. That, in turn, means your vehicle is running on borrowed time off the battery alone. You can keep on driving as long as the battery has enough juice left in it to keep the ignition system, fuel injectors, fuel pump and other electrical accessories working. But sooner or later, the battery will run down and your engine will die. How long this might take is anyone's guess. If you're driving during the daytime and aren't listening to the radio or running the air conditioner, the battery along might keep you going for four, five, six or more hours. On the other hand, if its nighttime and you have the lights on, the heater or air conditioner and radio, you might run the battery dead in an hour or two.
A warning light should not be ignored. It usually means something has failed and needs to be fixed -- the sooner the better in this case.
Sometimes the alternator warning light will come on, (or the amp or voltage gauge will show discharge) when nothing is really wrong with the charging system. It's a gauge or light problem. One way to tell is to turn on the headlights. If the headlights shine with normal intensity and do not change as you rev the engine, all's well with the charging system and the problem is in the warning light or gauge circuit. But if the lights are dim or get brighter as you rev the engine, you have a charging system problem that requires further diagnosis.
Any of a number of problems can cause the alternator warning lamp to come on. If the alternator drive belt is slipping, has broken or flipped itself off the pulley, the alternator will cease to produce current and the warning light will come on. If the regulator that controls the alternator's output has failed, the warning light will come on. And if there's a wiring problem in the charging circuit, the light will come on.