Short answer: No.
Long answer: The attorney general, or his designee, meaning an ATF agent, can require the owner of a NFA item (suppressor, short barreled rifle, short barreled shotgun, machinegun) to show proof of registration, i.e. the tax stamp. 26 U.S. Code § 5841(e).
There is nothing that requires a private range officer to check for a tax stamp. As a private actor, the range safety officer (RSO) may ask you to produce proof of registration as a condition for use of the range. The RSO may even prohibit you from using the range if you refuse. However, the RSO cannot compel you to produce the tax stamp. The RSO is not obligated to check for tax stamps any more than they are not obligated to check firearm serial numbers to ensure the weapon is not stolen. I am not sure where this myth started but it seems quite common. Let me say it again for emphasis: RSOs are not law enforcement agents and have no duty or right to demand an owner produce proof of registration.
I was at the (only) local indoor rifle range today when the RSO said he needed to see my tax stamp. I hate shooting there because they are on the whole a pretty unfriendly group. Anyway, I digress.
“It’s the law, I have to check” he said. I replied, “No, it is not. It may be your policy, which is your prerogative, but it is not the law and you are not required to check anything.” I reluctantly pulled up a copy of the stamp on my phone and went back to shooting. Had I refused, I would have been well within my rights. The RSO would have been within his rights to ask me to leave his range. I chose not to go that route.