Wow, that's a tough one. Epoxies are specifically made to be unreactive, permanent coatings and do a pretty good job.
Epoxies are polymers which contain multiple ether linkages with a few amine linkages. The resin is a polyether with active epoxide endgroups, while the hardener is a small multiamine. Once they react, you get a three dimensional crosslinked network that is mostly polyether with a few amine linkages holding the polyether chains together. Both of those functionalities are very stable. Depending on what is between the ether linkages, you might try a strong mineral acid like hydrobromic or hydroiodic acid.
In industry, most people count on the fact that depending on the exact epoxy, they might be able to swell the epoxy enough with a solvent to scrape it off without chemically breaking down the polymer. This is done using primarily dichloromethane, but also mixtures of solvents including DMF in acetone, Dichloromethane in methanol, and some even more complex patented and trade secret compositions (for example,
http://www.dynaloy.com/general-industrial/cured-epoxies. None of them actually break down the polymer, they just swell it and weaken it enough that it can be scraped away. I don't think this would be an effective way to free your carbon fibers.
Burning it might work, if you could keep the temperature below the ignition temperature of your carbon fibers. The nature of the fumes coming off would depend on the specific polymer and what sorts of molecules were between the ether linkages. These are usually highly aromatic though, so expect a lot of thick black smoke and toxic, possibly carcinogenic polyaromatic compounds.