If I create a character and attempt to destroy my hearthfire I can drain my entire stamina bar and nothing happens.  (I believe a similar thing happened with a personal claim, which I could not damage, the hour-glass never moved at all, using over 100 strength and a pickaxe.  However, I couldn't tell if that was due to lack of strength damage or if I was bugged at the time.  That character is unfortunately now dead.)

Explanation:
Someone else can make a character and destroy their own hearthfire immediately.

Someone else can make a character on my account, alongside the character that cannot destroy their own hearthfire and destroy their own hearthfire.

Someone else can log into the character that I created that cannot destroy its own hearthfire, and they too will not be able to destroy that character's hearthfire.

Someone else can make a character on my account, alongside the character that cannot destroy their own hearthfire, and I can login to that character and destroy their own hearthfire... however I can then login to the other character and it still will never be able to destroy its own hearthfire.

Schematic:
You can destroy the hearthfire immediately on your account.

I make character A.  A cannot destroy.
You make B.  You can make B destroy.
You cannot make A destroy.
You make C.  I can make C destroy.
Nobody can make A destroy.

An example account is "Hearthy" who has the character BeaterDown which cannot destroy its own hearthfire... all other permutations using that account were successful destroying their own hearthfires.