Does stoking the fire do anything once the builder has recovered?

Manually stoking the fire after you revive the builder has no functional impact on the game, so long as you never run out of wood.

The game checks the temperature of the room under two conditions:

• determining whether the builder should revive further
• when you try to build something (the room must be warmer than "cold" or the builder will just shiver)

The second option only happens if you run out of wood for the builder to use to stoke the fire. If this happens, you have to gather more wood, then either wait for the builder to stoke the fire, or stoke it yourself.