Most probably not. It's probably "Chinese" in the same way that a bunch of things in the US are called "French" that have zero connection to France. It only sounds exotic and it's fallen into slang as a nonsense attribution.

I looked up "Chinese burn" in the Oxford English Dictionary. It states that it's "children's slang" chiefly used in British, Canadian, and Australian English. The examples they give of the term date from 1956 to 1997.

As a kid in the US, we always described it as an "Indian burn." I'd never heard of a Chinese burn until about a year ago. The OED defines an Indian burn as the same thing as a Chinese burn but that the usage is chiefly used in American English. The examples they give of it date from 1954 to 1997.