Most people underestimate how broad the definition of irony is.
It is literally "results other than intended," which means that every example Alanis cited technically qualifies.
While dying from burning a symbol of something you hate is certainly ironic, Alanis Morissette doesn't seem to grasp the concept. Some of her examples do qualify, or would with some build-up, most just miss the mark. "A black fly in your Chardonnay", "Rain on your wedding day", "Traffic jam when you're already late"... and so on are clearly not ironic, but just unfortunate (but entirely reasonable and not at all "contrary to expectation"). Bad luck does not, by itself, constitute irony.
As for the original Greek meaning... well, hardly anyone sticks to that definition anymore.