If you take three 1.5 volt batteries you have, in principle, 4.5 volts. But the white LEDs want about 3.6 volts. That's 0.9 volts excess grunt.
The difference between the two batteries types is that Alkaline batteries sag in output voltage when they are loaded, while the Lithium ones don't. So the alkaline batteries tend to sag down towards the required 3.6 volts anyhow.
Sometimes the manufacturer puts a little resistor in the circuit to limit the current, but in calculating what value resistor to use he assumes that the battery voltage will sag alkaline-style.
So his calculations work OK with Alkaline cells, but are all wrong with Lithium cells. Excess heat, and short life.
This only applies to the older cheap and nasty LED headlamps. Some of those rely on the battery sagging enough quickly that no resistor is needed at all. Rough...
The more modern ones have a bit of electronics in there which limits the current more intelligently. The problem is that 'smart' costs $$, but at least you can put Lithium cells in them. For instance, the PT Quad will take Lithiums, and this headlamp is currently being tested at www.backpackgeartest.org . Caution - I am one of the testers!
The best answer is to use a full switch-mode converter inside the headlamp. However, the cost of the electronics would put the headlamp into the $60-100 category (at least), and this may be too much for the market for one of the little headlamps. It is OK when the headlamp has two 5W super-grunt white LEDs for illuminating half the mountain, but who needs that amount of light around camp? (Yeah, the military, I know ...)
The alternative is to make your own bit of electronics. High frequency stuff though. I do this, and power my 2-LED headlamps off a SINGLE lithium AAA cell. This is UL.