You only need to visit a forge when an item has been fully heated for its current level.
There is a forge in Haven located on the hill near the alchemy machine, and to the right of the fountain. This is a good place to perform your first forging, as there is an NPC at this location that offers some elaboration on how the forge works.
When you have an item that is ready to be leveled up, it will appear in an item selection menu. After selecting the item, a dialog box will appear prompting you to give a number of "fire crystals" in exchange for the level-up.
The forge features something of a gambling mechanic whereby you can offer more fire crystals in exchange for a chance to skip a level completely, get a heat bonus, or to obtain a Forge Prize box. This gambling mechanic is also useful if you want to guarantee a level-up, as there is a possibility that offering anything but the maximum (x3 the minimum, in my experience) amount of fire crystals will result in a failure. In the event of a failure your fire crystals will be removed, but your item will still be fully heated and ready for forging.
Below is a list of fire crystals that can be obtained, and their appropriate star rating:
☆☆☆☆☆ Cracked Fire Crystal
★☆☆☆☆ Dim Fire Crystal
★★☆☆☆ Warm Fire Crystal
★★★☆☆ Glowing Fire Crystal
★★★★☆ Shining Fire Crystal
★★★★★ Radiant Fire Crystal
These can be obtained in normal gameplay by breaking open boxes in the clockworks. Playing on a harder difficulty setting will increase the odds of getting fire crystals and other items of the new "rarities" class. I came across quite a few fire crystals playing Tier 3 on "Elite" difficulty, so unless you're really short on fire crystals there doesn't seem to be much point in offering up the minimum amount to the forge.