How can I clean burnt-on product build-up off of the plates of my
ceramic flat-iron hair straightener?

Since it is ceramic, it will stand quite aggressive abrasive cleaning
without damage. Damped, soapy wire wool should do the trick. Try a
test patch first just to make sure it?s OK. Do it in a sink under a
slowly running tap because the particles of iron go rusty and can
stain. Finally wash carefully in the sink unless you want rusty
highlights in your hair. And of course make sure water doesn?t get
into the electrics -- if it does, dry out for a few hours.

Best