The phrase is an idiom; it doesn't mean burning actual money, but rather spending it without thinking. Ever since she got that promotion, she's been acting as if she has money to burn.
Pro Tip 2/3
Casual Tone
Used in informal settings among friends or in relaxed conversations, not in formal or business contexts. Look at Mike in his designer outfit; he's got money to burn, doesn't he?
Pro Tip 3/3
Possible Negativity
It can imply criticism or envy when you think someone spends money unwisely or too easily. Buying another car? Someone has money to burn!