When I was in the Navy, a "Fire Watch" was set when there was hot work going on (e.g. welding etc.), usually set on the opposite side of a bulkhead where the hot work was being done, but sometimes in the direct vicinity as well. The Fire Watch had the appropriate gear/equipment to put out a fire if one occurred. I was a fire watch many times in my career.
This anon telling us he's pulling Fire Watch duty tells me he's on the lookout for trouble ("fire") and he's prepared to takes actions to mitigate it.