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Since the 9.09.2020 update the game won't start due to a Media Foundation error.
You can either:

add "--skip-logo" to launch options,

or use a custom build of proton (e. g. GloriousEggroll's one).

We don't support Steam Play on Linux officially yet, but we've received some positive feedback on ProtonDB.
You can install and play House Flipper on your Linux PC if you enforce the option through House Flipper's Steam properties.
The option you're looking for is called "Force the use of a specific Steam Play compatibility tool" (at the bottom of the properties window).

House Flipper Support > Linux support (Steam Play) > linux_force.png

If the option is missing

First make sure that Steam Play is enabled in settings. Verify settings in the "Steam Play" tab.

If that doesn't help check you can try enabling beta participation in the "Account" tab.