The 5th level spell Greater Restoration is available to Bards, Clerics, and Druids, and removes one level of exhaustion. The DMG includes the Potion of Vitality, a very rare potion that among other benefits removes exhaustion. Outside of that, resting is the only answer.
The DMG presents two rest variants on page 267. Neither of these affect exhaustion directly, but the Epic Heroism variant allows you to take a long rest in a single hour. This would speed up the process of removing exhaustion; note that it has considerably more effect on the game than just that.
There is one other option that might work. You could kill the barbarian and bring him back to life with the Revivify spell. I don't actually know whether dying removes conditions; I think it's up to the DM. Note: I asked a question about this here.
You could houserule that a single long rest removes exhaustion completely. This shouldn't break anything, because the frenzy barbarian is the only character that is affected by exhaustion on a regular basis.
An important caveat that should be made here is that exhaustion is meant to be an impactful condition that requires some investment to remove. Mearls has tweeted specifically about the frenzy barbarian. http://thesageadvice.wordpress.com/2014/09/16/primal-path-rage/ So you may want to think carefully before making it possible for the barbarian to not have to worry about exhaustion.