The current you feed the diode is in milliamps, the light output of the diode is rated in milliwatts. You have to put more milliamps into the diode to get 250mW out. For a 250mW diode, thats probably around 300mA. For a 405nm diode, you'd want around 4 volts.
Though like Jerry said, all diodes are different, so 300mA may be too much, or it might not even be a 250mW diode.
A 250mW laser diode gives 250mW of light output. With a <100% efficiency you'll need to put in more than 250mW.
What do you consider burning capable? Getting smoke out of electrical table or vaporizing it?
Even focused 30mW won't burn anything except maybe ultra thin black paper. You'd need at least 100mW to start melting thin plastic.
You'll really have to clarify "burn". 100mW will burn your skin if hold if you hold your hand long enough but it wont set wood on fire.
Bluefan was spot on. Their is a huge difference in "burning" electrical tape and burning wood to make a fire. There you'll need about 2000mW+.
FYI. My 10000mw laser sets wood on fire unfocused in about 3 seconds.