Oh, I imagine they have supplies to last a year.

I say open them after 100-200y, make it a fun time capsule, then turn them into museums -- "this is one of the places where we buried alive the biggest leeches in society after they constructed the most lavish palatial bunkers to hide in during a time of global turmoil!" -- you can show people just what excess wealth to the point of detachment from society will drive a person to do, and what society will do in turn to them.

If blocking the ventilation doesn't work, I can think of plenty of chemical and biological agents to introduce to a ventilation system that will produce the desired effect. Filters only catch so much, have a limited lifetime and will deteriorate, and can be hastened to deteriorate faster. I doubt they're going to fit a HEPA filter factory down there.

And as for the armed guards, if the world is bad enough that the billionaires have hidden away in their bunkers, I doubt it's a world where fiat currency has meaning anymore. About the only ways I could see armed guards being kept in line long-term would be to have their families all held hostage in the bunker as well, and/or explosive collars -- and the latter can be hacked, I'm certain. The former, I'm sure they could try, but then you're not relaxing in luxury in post-apocalypse bliss, you're running a massive underground prison where every guard hates the warden and wants to help the inmates escape and murder the warden. That doesn't sound easy or fun no matter how much money you have to throw at the problem.

Really, I find it unlikely that billionaires will be able to retain much in the way of long-term control if the outside world gets bad enough that they've sealed themselves inside a hole in the ground, no matter what kind of connectivity they have down there. They can build whatever they want now, while fiat currency still has meaning; but once it stops making sense to worship the dollar and the planet is on fire, they're just some moron in a hole in the ground with lots of supplies. They're a loot cache.

An inner chamber in a bunker that's only large enough for himself and his family is more of a residential scale rather than a commercial scale. He can let in one handyman at a time with relative safety. It's not like he has to bring a whole maintenance crew into his sanctum to change the washers on his bathroom sink.