It depends on how they have to be removed. In crate would be the safest way for both humans and dogs, but some people have the crates in hard-to-reach areas and can only access the doors to the crates. They may have the crates secured to the wall or floor or even to each other.
Let's talk about fire, since that is what I know. How or whether to remove an animal depends on where the fire started and what condition that area will be in if you have to carry your tools, hose and a dog crate through it, all while wearing turnouts and an air tank. If there is a window, an animal can be passed through to other personnel, but is the crate too big to fit through it? The dog will have to come out of the crate in that instance.
Leaving the animal in the crate and carrying the crate out is safest also because dogs bite when they are upset or afraid, they run away and hide in comfort spots where we might not be able to find or reach them…they are unpredictable in situations like that. However, once outside in the crate and the animal needs medical care, how difficult will it be for rescuers to get hands on the animal?
Firefighters do their best to rescue animals and having easy access is a great way to assist them doing this. That said, though, human life and safety must take precedent regardless of the screaming pet owner outside.