When some spoiled, self absorbed, rich douche is complaining about his or her trivial annoyances when they should have better things to worry about like the rest of us poor chumps... paying bills, buying food, etc.
Extends to rich business owners and corporations when any kind of rule or regulation is imposed on them.
Rich guy: "My new Italian Marble floor in my fifth bathroom has a crack in it."
You: "Sounds like a Luxury Problem."
The new regulations say I have to pay to clean up the toxic chemicals I dumped in the river! Now I might not be able to buy another vacation home!
Yup, Luxury Problem for sure.