My guess is that while the laser pointer is running, current is still flowing in the bulb. With an incandecent, this is OK if the current is low, as the fillament appears to be a dead short as long as the wire is cold. The voltage drop across the bulb will be almost zero volts, so the laser can have the entire 3V to play with.
If you use an LED, they light up on almost any current over 1-2mA. They also use a step-up converter, so they will present a high voltage drop. So the laser will not be able to get enough voltage to light up.
I doubt that anything would break if you tried it with an LED, but do not expect it to work.