It means "don't look forward to it". One more Ex: She said she'd get back to us, but don't hold your breath!

I would say it's closer to "Don't expect that to happen". So if you're hoping your best friend will visit you but they're very very busy, someone might tell you "Don't hold your breath", meaning "Don't expect your friend to come over".

It's basically like saying, "I wouldn't count on it."