Demonstrational smelting iron I'm planning an instructional demonstration of smelting. It seems you may be able to take rust (hematite, $\mathrm{Fe_2O_3}$) and carbon (perhaps from charcoal briquettes), place this in a crucible of ample size, light the carbon on fire, and the formed iron may melt and accumulate on the bottom of the crucible.

Yes, iron burns, everyone knows you can light steel wool.  I'm hoping that the interface above the iron will become encrusted with enough slag and ash that there won't be much of the formed iron burning, and what's specific about steel wool versus trying to light a spoon on fire is that the steel wool has a very high surface-to-mass ratio, enabling it to be heated well as well as provide ample mass transfer area.  

What are the problems with this demonstration, and what kinds of solutions exist for this?  For example, I foresee that I need to use lots of extra carbon to make sure that any iron formed doesn't reoxidize.
The biggest problem you will have is getting a high enough temperature, not the oxidation of the iron.

Putting the crucible in a charcoal furnace won't produce a high enough temperature to cause the reaction or the liquefy the iron: you need to have a crucible that can be driven to a much higher temperature. Normally this is done by blowing air (or, if you have it, oxygen) into the furnace. In practice what is often done is to mix the iron oxide and an excess of carbon in a single vessel: the carbon burns to create the temperature and also acts as the reducing agent to create the iron. This still requires a source to force air into the mixture. This appears to be how ancient craftsmen did the reaction, kicking off the iron age. 

For an account of how to do this see this blog describing a modern recreation of some of those ancient techniques.