No, the Achaemenid Persians did not have gunpowder (which wouldn't be invented until over a thousand years after the Battle of Thermopylae) and I have never encountered any source that would back the notion that they had some sort of hand bombs packed with some chemical substance (incendiaries and whatnot).

well black power used for 'explosives' originated in China around the tenth century and was used in fireworks and signals. So it's highly unlikely that there had been explosives in the battle of thermopylae, especially since Zack Snyder's 300 is a movie adaptation of a graphic novel, hence many elements may have been made up/exaggerated.

And widening the question: what was the first fire-throwing weapon/battle technique? Flaming arrows? Catapults?