As of September 20th, 2016 (Dentons) the answer is almost always No. Unless you are a encrypted message provider like Signal, Telegram, Whatsapp or Snapchat - these export controls are no longer a concern. Nearly every app uses HTTPS these days and it defeats the purpose of the U.S. Export Registry if every app needs to be registered.
The countries that are export restricted are people you don't normally want as customers; Syria, North Korea ect... and are typically restricted from using apps for more fundamental reasons - like not having cellphone towers or even the ability to buy a cellphone. It is common for backend services like Firebase and Google to blacklist export-restricted IP spaces, so they won't be able to connect even if they get the app.