Xenon Flash Bulb

I'd like to turn the flash from a disposable camera into a strobe light that flashes 240 times a second. Would the bulb be physically capable of this rate?

Yes, with some limits. The amount of energy/flash will determine its life...

If I recall correctly, anything flashing faster than 60 times per second, your not going to perceive as flashing. Also, xenon flash tubes are powered by a charged capacitor which dumps all at once.

Take a picture of a crt monitor at 1/30 of a second and in most cases you'll see the entire screen. 1/60 of a second will show the raster lines of the refresh rate.

The slr flash units that have variable frequency flashes have limited power equal to the overall capacitor capacity divided by the frequency for time duration.

Low ESR (equivalent series resistance) capacitors capable of charging rapidly (in other word, multilayers ceramic capacitor or mylar capacitor) would be required to reach the speed of flashing thought to be impossible with old school electrolyte capacitor normally used in many flashes in the camera.

Want it to flash fast? crank it to 600 - 800 Volts in series with inductor to manage amperage so it don't explode in one shot. In other word, high speed flash power supply design is not trivial.

Not to mention cheap flash tube will fail quickly due to the fact it uses soda lime glass (to keep price on tubes low) - flashing in less than 1/100 seconds is actually very abusive due to amperage draw in that timeframe.