Are war horses usually trained not to be skittish around fire? Setting is fantasy; Tang Dynasty China is a good approximation, but I'll take details from other times and places if you've got 'em. Can a horse be trained specifically to accept the use of a flaming weapon from her back? (Don't worry about the guy holding the weapon. It's magic. )
I'm interested in the response to this from those who have more expertise. From my limited experience of being around horses when my parents boarded a few of them in our barn when I was a child and then as a rider as an adult, it's not so much specifically fire they're afraid of. They're afraid of sudden flashing movements because their eyes work differently than ours. If memory serves (please, horse owners correct me on this), horses have no peripheral vision. That's why they startle easily. Plastic grocery bags on a trail blowing in the wind are a common thing that can spook horses, for example. You can train a horse to not be frightened of, say, plastic bags via desensitization training. So I'd imagine it could be done with fire or anything else as well. Correction: They have peripheral vision with two major blind spots, What they don't have is depth perception: http://www.lovehorsebackriding.com/horse-vision.html ETA: And of course any animal would be afraid of fire when they feel the heat. But they're not any more afraid of fire than anyone else, aside from the vision issue. edeited for clarity. can't write today.
Considering that real life horses could be acclimated to elephants, camels, and the clatter of armor (all of which normally spook them), I imagine you could train them to tolerate fire's presence by repeated exposure.
Even though I understand what you are looking for, if the setting is fantasy, can't you just have Horses trained anyway you want them to be? Having said that... I have very limited exposure to horses and we had this one horse named Ruffian (which is now the name of the Horse for one of my MCs, . I loved that horse he was a good boy) Anyways, Ruffian seemed pretty smart, and of the other horses didn't seem startled easily. In fact, when a pony escaped his pin, Ruffian helped me to corral him back in, seemingly taking commands. So I would assume they could but for my Fantasy settings I just have them trained or with their own personalities.
I grew up around horses, I did trick riding, rodeo... you name it, I've tried it from the back of a horse. Anyone who uses horses on a continual basis, like for a job or specific task puts a lot of time in with the horse. While this might not be traditional training, it certainly makes a difference in tough situations. Since you're using a dynasty as an approximation, I'd be remiss if I didn't point out to you that even horses used for archery where known for their temperament. Which is horse speak for they didn't get spooked. The last horse I rodeoed with, when she came to me she was a ball of nerves. You couldn't look at her crossways without the mare getting rattled. After years of working with her, I managed to "recondition" her so that she never spooked. I could walk between her legs, rattle a bag, shoot off a gun... nothing phased this horse. What I'm getting at is 1. This is your story, don't over think the small things, just make sure they are plausible, and 2. If this horse is used for something like war, archery, scouting, you can bet someone put the time in, and the horse wouldn't spook under extreme circumstances, especially if someone was competent and riding it.