Suuuure it can (slight sarchasm), but a 14500 can be charged to 4.25, its not a "3.7V" cell.
it seems there are 2 types of ways they will make these curcuits, to boost the alkaline or lithium cell at 1.7V max, and run at about 1.5W, or to handle multiple cells or rechargables. some of it is due to the available PARTS they are using. most of it due to the thing falls completly out of regulation and goes into direct drive (before 4.0V too). some of it due to the light design/metal isnt big enough to dissapate the heat.
if you MUST run a L1T or other lights that are similar (usually asking for a single 1.5v type cell) on a li-ion, do it intermittantally, they will run at more than 3W , and there is not enough heat removal capacity for the led.
or , toss a bit of resistance in, or dont fully charge the cell. its NOT designed for it, i have tested it both with a power supply and in reality within the flashlight, and it can run hot enough to ruin the led, and also burn the curcuit.
now with that said, I DO run my L1t with a 14500, but only intermittantally, i dont do a freewheeling runtime test, it does get hot internally, really warm to the hand. (nothing to lose)
and i Have burnt out leds, in less than 100 hour runtime from running them as hard as this configurations will. but i also will replace everything in it at will, when it dies.
umm, so conclusion, its your call, it WILL run it to hard, it WILL cause it to fail sooner, and the light will be replaced by something else , hopefully before you think its an issue.
ALSO, it runs the Battery above spec, the battery seems to handle it ok.
a protected battey has a bit more resistance than an unprotected, so it will be a bit safer. An unprotected cell will be deep discharged in ANY 1.5V type operating led light, and ruin the cell.
i still say , as i put in the compatability chart thread, its TO HIGH, so generally you would say NO, dont do it. use a lithium primary, it works ok.
surely you have a 15$ ampmeter, take off the end cap, and make the connection, do a little "watts" math | Volts X amps = watts |, and see it yourself.