It depends on who does the crafting.
The Forge of Thautam grants a feat without otherwise affecting how the feat works, so anything crafted on it behaves like normal crafting (with the exception of not needing the feat) and will have the CL of the character who crafted an item using the forge.
Having a caster level is a prerequisite to craft.
Some snippets from creating magic items:
Note that all items have prerequisites in their descriptions. These prerequisites must be met for the item to be created. Most of the time, they take the form of spells that must be known by the item’s creator (although access through another magic item or spellcaster is allowed).
While item creation costs are handled in detail below, note that normally the two primary factors are the caster level of the creator and the level of the spell or spells put into the item. A creator can create an item at a lower caster level than her own, but never lower than the minimum level needed to cast the needed spell. Using metamagic feats, a caster can place spells in items at a higher level than normal.
Creating magic armor has a special prerequisite: The creator’s caster level must be at least three times the enhancement bonus of the armor. If an item has both an enhancement bonus and a special ability, the higher of the two caster level requirements must be met.
Creating a magic weapon has a special prerequisite: The creator’s caster level must be at least three times the enhancement bonus of the weapon. If an item has both an enhancement bonus and a special ability the higher of the two caster level requirements must be met.
The forge grants the feat, but not a caster level.
As well as the feat in question:
A dwarf using this forge can create magic weapons and armor as if he had the Craft Magic Arms and Armor feat.
You can create any magic weapon, armor, or shield whose prerequisites you meet. Enhancing a weapon, suit of armor, or shield [...]
Crafting magical arms and armor requires, as a prerequisite, that the creator has a caster level equal to 3x the enhancement bonus of the armor/weapon or the requirement for a special ability, whichever is higher. The Forge of Thautam is granting a feat, but it's not changing anything about how the feat works, so all of the prerequisites still apply. A character with access to the Forge but who does not have a caster level will be unable to craft magical arms and armor as they won't be able to satisfy that prereq.
Since it doesn't modify the prerequisites or function of the feat, anything crafted on it will have the CL of the creator (or the CL of the creator modified through the various options defined in the crafting rules).