He probably does. Or at least I assume a furnished apartment is always available. Buying "property" (and I assume in Zuckerberg terms, that means a large home or a piece of land) implies a more substantial investment and possibly a commitment to spend a much larger percentage of his time in DC.

In DC you rent an apartment to spend time at the halls of power. You need someplace to sleep and keep your stuff. Not just lobbyists, but also senators and representatives do this.

You buy a home to entertain powerful guests.

The Conways famously purchased an 8 million dollar mansion during the Trump administration, IIRC. This stuff goes on all the time. If you pay attention to these stories, you can start to understand how power really works in the US.

Bill gates in his recent interview with WSJ said one of his past mistakes he regrets is not having a Microsoft office in DC sooner.

Jeff Bezos has owned an opulent mansion on a huge piece of land at the top of the hill in Georgetown for over a decade

He likely stays in a lot of hotels/airbnbs.

The risk of a hidden camera or microphone in an Airbnb/hotel would be reason enough to buy your own place, if you were a public company CEO with this much power.