Personally, I think doing 3-part staff on fire is a little corny... Some of the performances I've seen with it have so much energy it'd be inappropriate to distract from the performer with fire. Then again, maybe I'm just old fashioned that way!

Still, now I want to dig mine out and play with it!
Robnunchucks, you're thinking of "san-setsu-kon," which is basically the Japanese equivalent of san-jie-gun - each mean "three section staff". A simpler term for the tool you're thinking of (the shorter variety, yes?) is sanchaku - nunchaku is the Okinawan word for "two section staff".
Guiermo, if you have a wood staff it could work, depending on what kind of wood, and whether it's treated and/or painted. If it's chemically-treated wood, don't use it; the fire will make all the gunk in it seep out and it's really gross and possibly hazardous - I've seen it happen on a curtain rod we were using for testing. If it's only painted, you can sand it off and it'll be fine. If it's another material like rattan, I suggest getting another sample of it so you can test it before you construct a prop with it; most rattan martial arts tools are treated and probably won't be suitable for use with fire.
If you want extra protection, you can wrap any wicked section with aluminum foil tape - it's inexpensive and quite useful to have on hand! Again, if the wood is treated, the heat will cause the stuff to seep out and the foil tape won't stick. Apart from that, decide where you want the wicks and attach them as you would an ordinary staff.