JOR:
I’m not sure what you mean. If moral realism is false, then all moral beliefs are false
Well, that depends on the meta-ethical claim about the meaning of moral “beliefs†that happens to accompany the anti-realism. A few anti-realists — error theorists, like J.L. Mackie — believe that moral beliefs express claims about a non-existent property, and hence all moral beliefs are false. Most anti-realists, however, have some other theory of how moral expressions are supposed to work which hold that they don’t make any assertions at all, either truly or falsely — among them emotivists, expressionists, prescriptivists, etc. For these folks, “Murder is wrong†doesn’t state a truth, but it doesn’t state a falsehood, either; it’s not a statement, but rather another kind of speech-act, and your propensity to engage in that kind of speech-act is, strictly speaking, something other than a belief with propositional content.
(but not harmful or objectionable, since absent some standard of good, there is no harm or benefit).
That needn’t follow. You could be an antirealist about moral goodness without being an antirealist about all forms of goodness. (Say, prudential goodness I take it that this may be one way to gloss, e.g., Mandeville’s position.) In any case, if one is an antirealist about all forms of goodness, that doesn’t necessarily mean that she is going to withdraw claims about what’s good and what’s not good; it’s just that she has a different theory about what it comes to, logically, when she makes a claim like “The belief in the curse of destruction is harmful.†On a realist theory, “harm†refers to some form of evil (moral, prudential, or otherwise) being inflicted; so the statement simply ascribes the property of producing those evils to the belief. On the kind of theories that typically accompany antirealism, the claim is still going to be made; it’s just going to be held to boil down to something like “Boo! belief in the curse of destruction!â€