Preventing the Recognition of Terrorist States Act of 2021 This bill imposes sanctions related to the Taliban and addresses other related issues. The President must impose sanctions on foreign individuals and entities that knowingly provide significant support to the Taliban or senior Taliban members. The bill also repeals an exception to existing sanctions against certain entities related to energy, shipping, and shipbuilding in Iran. Specifically, the bill repeals a provision authorizing the President to exempt certain entities involved in the reconstruction of Afghanistan from such sanctions. No federal department or agency may take any action that states or implies recognition of the Taliban's claim of sovereignty over Afghanistan. The bill also bars the Department of State, the U.S. Agency for International Development (USAID), and the Department of Defense from using federal funds to prepare or implement any policy that extends diplomatic recognition to the Taliban-controlled government in Afghanistan. The State Department must designate (1) the government of Afghanistan as a state sponsor of terrorism, and (2) the Taliban as a foreign terrorist organization. Furthermore, USAID must take appropriate steps to ensure that certain foreign assistance provided in or for certain countries, including Afghanistan and Pakistan, is not going to or through any individual or entity involved in terrorist activity. The bill also prohibits using certain federal funds to provide direct assistance to any country where the duly elected head of government has been deposed in a coup or decree in which the military played a decisive role.