One of the supposed justifications for the intellectual monopoly called copyright is that it drives creativity and culture. In the last few weeks alone we have had multiple demonstrations of why the opposite is true: copyright destroys culture, and not by accident, but wilfully. For example, the MTVNews.com site, along with its sister site CMT.com, …
On itself, a simple claim (like “copyright destroys culture”) cannot be fallacious. It can be only true or false. For a fallacy, you need a reasoning flaw.
Also note that, even if you find a fallacy behind a conclusion, that is not enough grounds to claim that the conclusion is false. A non-fallacious argument with true premises yields a true conclusion, but a fallacious one may yield true or false conclusions.
The issue that you’re noticing with the title is not one of logic, but one of implicature due to the aspect of the verb. “X destroys Y” implies that, every time that X happens, Y gets destroyed; while “X [is] destroying Y” implies that this is only happening now.