The finer points of this project elude me. Would it not just be a hole in the ground with some rocks stacked around it? If, for some inexplicable reason, a man wanted a fire pit in his back yard, wouldn't the whole basic pit/fire thing be the appeal, which would call for natural materials rather than concrete which is contrary to the primitive aesthetic one would seek to evoke in one's fire pit?
But if you're casting in concrete, plywood is the usual choice for a form. Layers of thin stuff glued/screwed together makes curves. Support forms strongly. Diesel is a cheap bondbreaker. Also heard of used motor oil, but it's carcinogenic. Knocking the form when the crete is wet will displace air bubbles and give you a finished surface. Idk if you'd have to use a refractory mix. Idk what to do about reinforcement bar either, since I think steel moves .001" per foot of length per degree F, so it might cause cracking rather than prevent it. Concrete seems like an ugly and unnecessarily difficult and expensive material choice to me when there are easier, free, and more attractive methods.
Someone once warned me not to use river stones in heat because if water in the rock turns into steam and can't vent in time then the rock will blow up, which sounded like a .00001% possibility to me but there it is, I've repeated it anyhow.