In case after terrorism case, from the Fort Hood, Tex., shootings to the Boston Marathon bombing and now to the slaughter in San Bernardino, Calif., the inflammatory videos and bomb-making instructions of Anwar al-Awlaki, easily accessible on the Internet, have turned up as a powerful influence. On Friday, the Counter Extremism Project, an advocacy group based in Washington, called on YouTube and other platforms to permanently ban Mr. Awlaki’s material, including his early, mainstream lectures. “His work has inspired countless plots and attacks,” said Mark D. Wallace, a former diplomat and homeland security official who is the project’s chief executive. “It’s hate speech. It should come down, period. Like child porn, it should be expeditiously removed.”