Your lamp timer probably has ran out, or the lamp is just deffective. Some projectors have a way to bypass the self-shutdown in order to allow the user to check/reset lamp timer. If timer gets near to the factory-defined lifespan (2000 hours for example), even if the lamp is ok the projector software will refuse to turn it on due to risk of cracking of the material thus avoiding physical damage to the projector.
As you said it runs for shorter than a minute, it could not be a heat issue. Notice that if you turn of the projector (when normal operation) you got to wait between 30 minutes to an hour before you turn it back on. The projector has a procedure of igniting the lamp that requires it to be ambient-temperature during the power up. Turning the projector back on right before turned off will dastically shorten the life of the lamp.
So, I'd say you'd better try changing the bulb. Refer to the manual of your projector, there must be a section dedicated to changing the bulb. On the troubleshooting section also should have tips about what to do in this situation.
Hope it helps!
avenger, May 2009