What is this smoking item in Beyond the Boundary? This is set on one of the tables in episode 9/10 and seems to burn inwards, pieces falling off as it goes - creating the spiraled chunks on the plate.

I've never seen anything like this before, is it something that exists outside of the show and what is its purpose?
This is what is commonly known as a "mosquito coil." These are mosquito-repelling incense, in spiral-shaped (this particular shape was popularized in Japan) and made up of a dried paste of pyrethrum powder.
I japan it's called "katori senkou" and it's usually green-ish swirly incense-like item, that you set on light and burn slowly. The smoke/smell that comes from it keeps the mosquito away.

As you may or may not know, just about all chemical repellents and insecticides contain pyrethroid, which is a chemical imitation of Pyrethrin, found in pyrethrum.