Yes, definitely 'wood fire'. Like 'coal fire', 'gas fire' and so on. 'Wooden' is an adjective referring to the material that something is constructed from e.g. wooden table, wooden chair and so on. You might have a wooden fireplace (meaning the wooden surround and mantlepiece), but you could not have a wooden fire. The chimney is the tall, narrow vertical ventilation structure which takes fumes and smoke from the fireplace upwards and out of the top of the house. If you get a fire inside that, you're in trouble!

Wood fire is correct. "Wooden fire" doesn't make sense. Wooden is an adjective and it can be used to describe a thing. For example: Wooden chair, wooden staircase, wooden shelf, etc. "Chimney fire" is incorrect because it is a situation in which chimney catches fire, it can't be used here since we're talking about a routine situation where there's always a fire. Chimney fire would mean an emergency situation.