^Beat me to it.
I had to look up the datasheet to figure out what I assumed Dave already knew.
"linear current sink."
Ok, next question.
So how does PWM help?
Is it possible that 1 amp at 10% duty cycle can be visibly brighter to a human being than 100% duty cycle at 100mA? I actually think this can be the case, given the properties of the LED, but perhaps only with very minimal current that is not close to the LED's max rating. I am thinking that at some point near the knee that doubling the current might possibly more than double the light output? (Of course near the max rating this is for sure the reverse... and PWM would be less efficient in terms of light output). Or maybe that strobed light is more efficiently used by the human eye? I dunno, but I was open to hearing all about it. I was wondering if this is what batteksystem was going to get at. Or maybe he just didn't realize his LED driver of choice is a linear current sink or that PWM in general wastes just as much energy for the given amount of current. W/e the voltage drop you are getting rid of linearly, it obviously doesn't matter if it's 100% duty cycle at X amps or 10% duty cycle at 10X amps... it's the same.
As far as I know, the most direct and efficient way to reduce that waste is to select the power supply to produce as little overhead as possible. Just enough to provide a fairly stable/constant current over operating temp range. By arranging the LEDs, selecting the appropriate power supply, and/or using DC converter in order to convert some of that excess overhead voltage into more current at a lower voltage. You will still want a series resistor, but a smaller one that is dropping less voltage.
Using a DC converter WITH constant current output may or may not be as efficient at any given power supply voltage, but it will be more stable. The active current limiting/adjusting part is a red herring that doesn't actually increase efficiency. It's the buck converter (and boost converter, if you're talking about extracting more energy from sagging battery output; so yeah, from a battery, this sort of switchmode IC with current regulation will certainly be king) where you reclaim that wasted energy. Essentially, an LED uses only X volts. If you don't want to waste power, feed it with as close to X as possible to give the desired results.
OP, to keep it simple: If FVD is 3.0V, and you can provide 3.1V with a small series resistor to drop the excess 0.1V, AND IF the brightness/current draw over the temperature range and battery life is acceptable... that will waste only 1/10th of what happens when you use a 4.1V power source. The tradeoff is your brightness/current draw curve is going to not be as flat, considering battery output curve.