First, let me give the disclaimer that I am not aware exactly what the suppressor regs say. I don't own one, or a full auto arm and am not a lawyer, so my advice is worth what you're paying for it.
My
understanding is that there is a required "compliance inspection" for certain NFA items. This is a check to see that you still have the item their paperwork says you have, AND that you have the paperwork on it (tax stamp, etc) that you are required to have. It might be yearly, I don't know.
This is a scheduled meeting between you and the agent(s) doing the check. They will make an appointment with you, to coordinate the time and place.
Now, technically, they will be visiting you without a warrant. THIS IS ALLOWED, as you are inviting them, they don't need a warrant.
They are not allowed to show up unannounced, in the middle of the night, or any other time, and ransack your property, its not a criminal investigation (for which they would need a search warrant, or your permission) its an administrative check, and you do have the right to refuse them entry to your property.
You will have to produce the item and your paperwork, but it doesn't have to be in your home, if you don't want it to be.
You're not giving up your rights when you get an NFA item, but you are required to play by the ATF's rules, and you are agreeing to do so as part of the application process.
If I've got any of this wrong, please feel free to correct and enlighten me.