Wire brush, the kind used to clean outdoor grill. I wonder what would dissolve plastic? I'll see if I can find something on Google. Probably pretty harsh stuff.

Found a few. One, nail polish remover. Another, not sure about, most suggestions were for removing plastic from pans. A paste of baking soda.

If you Google removing burnt on plastics there a a bunch suggestions.

I remember trying to put paint thinner in a styrofoam cup many years ago and the cup dissolved in my hands. I don't know what that grate is in a fireplace? If removable to outside, I might try paint thinner, but if that's indoors, I would not, esp if you have a gas fireplace. It's not the screen, but a grate? If you can take it outside, maybe a butane or propane torch?

Place the grate over a fire to char the plastic, mechanically remove the char, then repaint with hi-temp paint.