The Lunar Forge works no differently that other normal forges.
The Lunar Forge is located at Silent Moons Camp northwest of Whiterun. On a Workbench near the forge is a book called Notes on the Lunar Forge, which suggests that weapons smithed here will convey a "lunar" damage bonus at night when the moons are out. There are several random Lunar weapons around the forge, which do have these properties. However, the Lunar Forge cannot actually be used to create new Lunar weapons. Instead, "Silent Moons Enchant" is an enchantment that can be learned by disenchanting one of the Lunar weapons, and can then be applied to any weapon at any Arcane Enchanter.
The lunar weapons found near the forge have a Silent Moons enchantment. Those don't work too, due to a bug.
The Silent Moons Enchant is not actually controlled by phases of the moon. Analysis of the game files reveals the enchantment is designed
to deal damage between 9pm and 5am, regardless of lunar phase,
weather, or indoor/outdoor location. Note however that due to the bug
listed below, the enchantment does not actually deal damage even
during allotted times.
The Silent Moons Enchant is broken. While targets hit by a Lunar-enchanted weapon will glow as if the effect were working, PC
testing via developer console confirms that no extra damage is done.
Analysis of the game files reveals this to be due to the enchantment
targeting weapon speed and not health.
This bug can be fixed by the Unofficial Skyrim Patch:
The health damage effect of the Silent Moons enchantment of Lunar weapons will now work properly.
Full changelog: http://www.iguanadons.net/Unofficial%20Skyrim%20Patch%20Version%20History.html