I think Nikongod has had some sticky headphones.

But what did you finish the headphone stand with? The finish is probably what's doing it. Some wood finishes are solvent based or have solvents in them. And even when the finish seems dry to the touch, it can take weeks or months to fully cure depending on the weather and the moisture content of the wood.
First, I'd wipe the headband down with isopropyl alcohol. Not rubbing alcohol, which is fairly dilute. Hit Radio Shack and see if they have any for PCB work. That should get any residue off. Denatured alcohol should work as well, and you can get that at hardware stores.
As for the stand, would you consider stripping and refinishing it? I know you probably worked hard on it, but you don't want it to eat the headband. I'd recommend using only wax as a finish. The stand will hold up fine with wax, and you'd need to apply more every 6-12 months.
Another option is putting a metal plate over the rest. Copper is easy to work and looks good. Worth considering.