Owning NFA items such as suppressors and machine guns can be a tricky proposition. They're expensive, they're subject to a $200 tax and a long approval process, and they're illegal in Georgia unless you can prove otherwise.
Both the Form 4 and the ATF's own opinion imply that proof of registration must be provided to an officer of the ATF upon request. No other parties are mentioned, so many owners assume the ATF are the only folks who have a right to demand it.
However, most states have their own spin on this. According to Georgia Code 16-11-122:
No person shall have in his possession any sawed-off shotgun, sawed-off rifle, machine gun, dangerous weapon, or silencer except as provided in Code Section 16-11-124.
The subsequent section levies a 5-year prison sentence for owning those items, and 16-11-124 provides an exemption if the weapon is registered "in accordance with the dictates of the National Firearms Act." However, 16-22-125 states:
In any complaint, accusation, or indictment and in any action or proceeding brought for the enforcement of this part it shall not be necessary to negative any exception, excuse, proviso, or exemption contained in this part, and the burden of proof of any such exception, excuse, proviso, or exemption shall be upon the defendant.
This makes the Form 4 an affirmative defense. You are presumed guilty, and it's your job to prove that you are immune from prosecution. In this case, that would entail furnishing proof of registration.
So yes, it would be a good idea to keep a copy handy, and it would be a good idea to provide it to law enforcement when asked. The alternative could be vastly inconvenient and costly. Range officers may also ask for proof as well, as it's one of their responsibilities to ensure that illegal activity isn't taking place on their watch.