Nothing. They stay there. However, since you are the claim owner, you can remove them at any time with no penalties as long as they remain in your claim.
Opinions expressed in this statement are the authors alone and in no way reflect on the game development values of the actual developers.
It is worth removing them sooner than later due to the danger of having random hearths in your home. They can spawn other hearths in the area, steal and destroy things not on claimed land, and generally be a nuisance. Since it doesn't require criminal action to remove them it is well worth it.
you'll need 25 strength and a stone axe to damage the hearth fires(suprised neither of those 2 mentioned it they're usually super helpful). a bear cape, or a glass of wine will give you a strength boost to help you out early on with those.