Hanging from, could show that the item is being held or hanged on one end by something in this case, a chair. “Hanging on” in this case seems most appropriate.

In my opinion, both of those phrases could be used interchangeably. However, I feel that saying “a jacket is hanging from a chair” would be used in a more descriptive sense, like in a book or a story where the author is trying to describe the room, whereas “a jacket is hanging on a chair” would be used more in everyday conversation, for example if you want to tell someone where the jacket is.