Yes and no.

You are not required to show your registration form (which is a tax document) to anyone except an ATF agent or other designee authorized by the head of the ATF.

However, most states specifically outlaw machineguns, unless possessed in accordance with federal law (meaning, registered properly under the NFA). Failure to show that tax document could result in Deputy Barney seizing your NFA toy, and arresting you for illegal possession of the same, unless and until you can prove otherwise. You're not "innocent until proven guilty" when something is illegal except for an affirmative defense, and you don't provide that affirmative defense.

In states which do not outlaw mg's, John Law still probably doesn't know the rules, and might arrest you, but you might have a chance of suing for wrongful arrest after you get out of the hoosegow.

It's a safe and smart practice to keep copies of your registrations with your toys, present them to law enforcement if they want to see them, and do your best to be polite and educate them about NFA laws.