The secret to disassembling ferrite cores, transformers, etc.:
Heat and superglue.
(Not at the same time.)
If the transformer is small, you can heat it with the soldering iron. DO NOT apply direct heat to the core. The core is weak ceramic and will crack from thermal stress!
First, remove any tape or clips securing the core.
Set the soldering iron horizontal, and balance the transformer on the barrel of the iron. It will heat up gently this way. Using gloves and/or pliers, twist and wiggle the core pieces until they come loose. The varnish should become gooey (and a bit stinky) when it's hot enough.
Larger transformers will take a long time to heat up, and may be better done in a toaster oven (you DO have a throwaway toaster oven for lab use only, right?).
The core may end up fractured, anyway. Clean the fragments: scrape off whatever excess varnish you can, and wipe the fractured surfaces with solvent. Line up the fractured surfaces as perfectly as possible, then mate them with a drop of superglue. Hold until cured.
The additional air-gap of a thin superglued joint is insignificant for most cores. (High-mu pulse transformers and common mode chokes are the most sensitive to air gap.) If medium sized chips are missing from the fracture, try to reassemble them; don't worry about small chips. Large chips, and crushed material, isn't practical to repair.
Inevitably, the core pieces won't line up perfectly, and the ground faces will no longer be flat and parallel; if this is desired, use carbide sandpaper (laid over a very flat surface, like plate glass) to grind the faces parallel again.
Tape and wire can be unwound from the bobbin. Be careful with the bobbin: it is made from brittle, high temperature plastic. Salvaged wire is sometimes reusable, but don't count on it. (Some varnish is so strong, it peels the enamel right off the wire!)
Do invest in a roll of Scotch(R) 74 or similar polyester tape (it's usually yellow). It's legit, it has good hold strength (better after heating!), it's strong and it's an excellent insulator.
Or polyimide (Kapton(R)) tape, which is even stronger, stiffer, and higher temperature (and more expensive). Though I'd kind of recommend against it for general use, as it's not as well-behaved (because of the stiffness and all that). It's good for other uses too, though -- it can withstand direct soldering heat pretty much forever. Polyimide is one badass organic polymer!
Tim