Lots of good info in this thread already. As a budding bladesmith and long-time DM, let me add a little:
Smelting/Bloomery operation: You do need high temperatures for this - this is the stage where you are actually melting the iron, separating it from the rock and impurities it is contained within. Of course, if your world is set up for having large veins of pure metals running through it (not realistic, but fine for an rpg) then this step is not needed.
Forging: Decently high temperature needed, but nothing close to what you need for the previous step. With a proper forced-air fire and any decent fuel you should be able to achieve the proper environment. (Some cultures have used long blowpipes or channels on a windy area to force air into the center of the fire - either of these is available to dwarven engineers, and probably more intricately made
) Even welding temperatures (forge-welding two different pieces of metal into one) aren't as high as smelting temperatures.
Heat Treating/Tempering: The final steps in the actual forging process, these involve heating the steel to a specific temperature, then cooling it at a controlled rate (this is the 'quenching' process, where the hot blade is thrust into a vat of oil/water/brine, etc). This process is quite complicated, but for the current discussion about heat what you need is a controlled environment. A good smith can use his regular forge and judge by eye (quite within the means of a good dwarven craftsman, I'm sure). You don't need temperatures out of the range of standard forging.
How could these temperatures be reached without coal in an underground environment? I would say that the fuel is important, but not the whole picture. You would also need a good constant blast of air to the forge (as discussed above, easily within reach of the average dwarven engineer's skill). A good refractory material would help tremendously as well. A refractory reflects heat, trapping it in the forge, helping to reach higher temperatures and save fuel. Perhaps the dwarves found out long ago that a certain paste of minerals and special underdark mud reflects heat well and began using it in their forges.
The above is all mostly concerning steel and ironwork. Other metals like bronze, copper, etc forge at lower temperatures. Adamantine and other metals probably forge at higher temperatures, or perhaps they only achieve their full hardness from a special hardening/tempering process, and forge at steel-like temperatures. That's really up to DM fiat, as the DMG doesn't go into forging temperatures for special materials
Regarding your dwarves and their coal shortage - if that is a plot point, if you want them to be low on fuel for forging, then that's fine. If you just want them to be low on coal, they could find any number of alternate fuels. Peat has been mentioned and could work. If there is a source of wood nearby, they could simply use charcoal (it works very nicely for forging). Lots of opportunity for adventure there, as they would have to venture out of their caves, cut trees, and burn it into charcoal all on the surface somewhere (unless you have underground trees, or mushrooms with a thick, wood-like structure to them).
