Depending on the specific place and state laws. The example of a furniture store is a fair one, if we contrast it with a bar/Tavern, or a bank.
State laws prohibit carry of arms in specific private places (like those serving alcohol, or a bank), just as they do certain public places. However, they do not generally bar the carry of arms on other private property, like a furniture store, or a mall, or the 40 acres of woodlot owned by a paper company, or the farmer down the road.
In those privately owned places not specified in law (open to the public, or not), a "no guns" sign means nothing, legally. It means something to the owner, and gives them a reason to order you off their property, if they desire (not that they need one, legally) but it carries no force of law.
What I mean is, that you cannot be charged with carrying a gun in Homer's Furniture, absent specific state or local ordinance against carrying.
Homer can tell you to leave, because he doesn't want guns in his place, that's his right. But he can't have the law after you for having a gun in defiance of his sign.
However, any action other than immediately leaving when Homer says so (including arguing about our Constitutional rights
) makes you guilty of trespass, and Homer can sic the law on you for that, and likely will.
Just as you can be charged with trespass when crossing "posted" land. There is in law a process for posting land against trespass, but there is none that I am familiar with concerning a "no gun" sign. At least, so far.
But, as pointed out, different locals may have differing laws, so check yours carefully.