Hopefully someone with actual legal expertise can chime in since IANAL, but…
My understanding is that signs are not enforceable in the same way that they are in other states like Texas with their 30.06 and 30.07 signs. In those states the signs must be displayed in a specific way (size, text, location), gun owners know where to look for them, and if you are caught with a firearm then just being there subjects you to the penalty (in Texas their 30.06 & 30.07 signs result in misdemeanors and fines, other states may vary).
In Virginia, there is no “official sign” with rules on where it is to be displayed. If a business has a “no weapons” policy of some sort, and they discover you have a firearm they can ask you to leave because of that policy. As long as you leave right away there is no problem. But if you don’t leave then you are trespassing and you would be subject to any of the penalties that go along with that. I am not aware of any “extra” penalties for trespassing with a firearm where it is banned by the property owner.
EDIT: I don’t think I’ve ever seen a “no firearms” sign. I have seen some “no weapons” signs I think at doctor offices and they were big right at the door. Most other places you’re just supposed to “know” you can’t carry there like courthouses, schools, etc