Because it's made with a few different materials, you might not think of a 3-ring binder as a good candidate for recycling. But after a simple disassembly, you can reuse any of the parts for a new purpose and recycle the rest.
Cut or slice the plastic covers from the binder's spine piece, using a utility knife. Slit one end of each plastic cover, and slide out the cardboard insert from within.
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Use pliers to separate the spine piece from the ring mechanism. Unless you can think of a use for the spine, don't worry about damaging it during removal.
Keep the cardboard cover inserts for art projects or as stiffening material for mailings. If the plastic covers are clear, you can use them as paper protectors, such as for filers for event-table displays: put a flier inside the cover (to keep it clean and weight it down), and store extra fliers in the pocket on the flip side.
Recycle any materials you won't use. Sort cardboard with papers; plastic with plastic bags; metal with cans.