It'll definitely reach equilibrium eventually, that equilibrium might not be tolerable though. Like you said, water pipes are probably the most efficient way to cool the underground chambers, but you absolutely have to have air flowing down into them too, and ventilation carrying away harmful gasses from the smelter operations. One way dwarves could do that is to design a cooling tower built into the mountain, exactly the shape of modern nuclear reactor cooling towers, built either underground pits exiting onto the surface or as towers above ground. That will pull hot steam and air out, allowing cooler air to be drawn in through inlet vents, providing circulation. Magma forges are where it gets really icky, because you have not only the noxious gasses produced in normal smelting operations, but the noxious gasses naturally found in lava. Dwarven engineers would absolutely have to use separate ventilation systems to pump those gasses out, keeping them downwind of the fresh air inlets as some of them can be quite dense and sink back down into the fortress.
Keeping the magma flowing is one issue for them, but they would need to prevent pressurized magma pockets from bursting with all the force of a volcanic eruption, turning the fort into a new volcano pipe. It's under constant pressure from the rock and magma above and around it, and any empty spaces it can burst into it will, killing everyone inside and filling the chamber until pressure is equal throughout. Even if you were trapped in an air pocket inside there, you'd be experiencing pressures akin to the deep ocean or worse, where you'd need a submarine structure to prevent you being crushed to paste. Dwarves are not immune to being crushed to death, unfortunately.
You can use that pressure to your advantage to allow hot, fresh magma flowing into your smelter, but where is it flowing to? You can't pump it back down into the magma sea, you just get another opening ita trying to escape to and now your pipe has two high pressure ends and a blocked center. Dwarf fortress has giant open underground caverns, you can let it flow into the third cavern layer, but dwarves don't want to lose access to a very precious resource there; nether cap trees, whose temperature is (somehow) constantly quite cool and act as an eternal heat sink. Building nether cap paneling into every surface of the lower forts or cooling systems would solve the overheating issue.
Dwarves have access to basic mechanical systems to pump air and water into the fort below (but probably not sufficient to do the job irl), but not really anything to stop magma from bursting up through the floor. Unless the magma is no longer under high pressure for some reason, which in dwarf fortress at least is the case, as all the open lava tubes to the surface show. Whatever level the lava tube fills at, it will remain constant, without the entire magma sea attempting to thrust skyward as the solid earth collapses on it. If that's the case, the weight of the earth above the sea must be supported by something other than the magma sea. Probably the columns of adamantine, I imagine supported by networks of fine filaments between them, holding the world up like a great hammock.