I'm presuming that the shiny metal is stainless steel, or has a finish that won't be damaged by solvents. It's hard to know what to suggest, other than trying a scrubbing pad and cleanser, or possibly a solvent like lacquer thinner, naptha, or MEK. If it were me, I'd call the manufacturer and get their advice as they will know what the metal is and what the plastic is. I also expect that you may be able to destroy/vaporize/carbonize the plastic just by using the grill on high.

I agree, eventually with bad smell/pollution, it will burn off, but if it's not totally baked on, a solvent like wd-40 or diesel will remove adhesive, but you must of course clean it thoroughly so no traces remain before you fire it up.

I can almost guarantee it's stainless steel which commonly does come with a protective plastic that's applied whilst it's still sheet metal, then they stamp/cut it with the film still on.

I would place some cardboard inside the grill and get a good fire going inside to burn this off. Then maybe use a wire brush to remove any remaining crud.

I would not cook anything until all the plastic is completely burned/cleaned off.

The only other way I know of removing such plastic is via a large vat of hot solvent (an extractor). I used to use one of these on a job I worked on a while back. The extractor would probably remove all your outside paint and finish as well.

Is there any kind of finish inside the top or just bare metal?

There are a number of solvents that should be able to attack and remove the plastic residue, some more dangerous than others if the surface has no finish (bare metal).

If you use the correct solvent it is very unlikely you would need heat, just tincture of time.

DO NOT burn cardboard or wood or anything else in it that's not designed to go in there, you will ruin the finish, the handles and other fittings. A bad idea.