It has been said that if there is human freedom, then we are responsible for our actions. By this, it seems natural to suppose that "given that there is no human freedom (let's just suppose for the sake of argument) then it would follow that we are not responsible for our actions." But this seems an instance of what is called the "fallacy of denying the antecedent". Is this really an instance of the fallacy or is it an exemption to the case because personally I don't see any error in the form of the argument.

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Translating the argument into symbolic terms quite literally, we get this: 'If F, then R. Not F. Therefore, not R.' That form of argument does indeed commit the fallacy of denying the antecedent: the premises don't logically imply the conclusion; the truth of the premises doesn't logically ensure the truth of the conclusion. The first premise says that F is sufficient for R; it doesn't say that F is necessary for R. In that case, R can obtain even if F fails to obtain.

My hunch is that you're interpreting 'If F, then R' as 'R if and only if F': you're interpreting the conditional as a biconditional, i.e., as the claim that F is both necessary and sufficient for R. 'R if and only if F' and 'Not F' together imply 'Not R'. Your interpretation is understandable, because conversationally we often do intend to assert a biconditional when we use conditional language. A parent's 'If you clean your room, you can watch TV' usually means 'You can watch TV if and only if you clean your room'. But on a literal interpretation the argument you offered is invalid.

In the form you've presented the claims, there would be a fallacy of denying the antecedent. If free, then responsible. Not free. So, not responsible.

But I don't think philosophers typically agree with the conditional claim, which says that having free will (or doing A freely) is sufficient for moral responsibility (or being responsible for A). And we should not agree with it. After all, I might freely decide to back my car out of the driveway and in doing so run over the sleeping cat I could not be expected to have seen. If so, I do not seem to be responsible (blameworthy) for killing the cat. There might be ways to fix up the terms, but there is likely an epistemic condition (a justified belief requirement) for responsibility that goes beyond the free will (or control) condition.

However, it is more plausible to say that moral responsibility (being responsible for A) requires free will (that one did A freely, or did some earlier action freely that one should have known would lead to A). So, if I am responsible for killing the cat, I must have free will and must have exercised it in such a way that led to my cat killing.

Suppose we accept: If one is responsible, then one has free will.

Then, by modus tollens (or denying the consequent--i.e., saying we lack free will), we validly conclude that one in not responsible.

Some people suggest that it is so implausible (and/or costly) to assert that humans are never ever responsible for anything at all (e.g., that no one deserves blame for anything) that we have good reason to question any argument (or premises) that concludes that no one has free will.

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