Transformer voltage step-up and heat
If your soldering iron heater is intended for 240v, then giving it 12v would result in a very low heat output indeed. Feeding it with 1/20th of the voltage would result in 1/400th of the rated power (\$power=\frac{v^2}{R}\$), you might feel the chill taken off it, but no heat to speak of.

Before you look at 24v irons made by Weller, they run off 24v AC. I doubt the thermostatic switch is rated for DC, so you could expect the switch to fail in short order if you tried to run on off 24v DC.