Argument A for the conclusion C is fallacious.
Like anything else, the concept of logical fallacy can be misunderstood and misused, and can even become a source of fallacious reasoning. To say that an argument is fallacious is to claim that there is no sufficiently strong logical connection between the premisses and the conclusion. This says nothing about the truth-value of the conclusion, so it is unwarranted to conclude that a proposition is false simply because some argument for it is fallacious.
It's easy to come up with fallacious arguments for any proposition, whatever its truth-value. What's hard is to find a cogent argument for a proposition, even when it's true. For example, it is now believed by mathematicians that the proposition known as "Fermat's last theorem" is true, yet it took over three centuries for anyone to prove it. In the meantime, many invalid arguments were presented for it.
It is reasonable to, at least provisionally, reject an improbable proposition for which no adequate evidence has been presented. So, if you can show that all of the common arguments for a certain proposition are fallacious, and the burden of proof is on the proposition's proponents, then you do not commit this fallacy by rejecting that proposition. Rather, the fallacy is committed when you jump to the conclusion that just because one argument for it is fallacious, no cogent argument for it can exist.
David Hackett Fischer, Historians' Fallacies: Toward a Logic of Historical Thought (Harper & Row, 1970), pp. 305-306.
Acknowledgment: Thanks to Adrian Larson.