The OEM spare tire carrier on my NAS truck rattles against the third brake light carrier. The problem worsens the larger the spare you throw on there, and the real fix is buying an aftermarket carrier that solves the rattle and, more importantly, takes the weight of that spare off the rear door.
In the meantime though, here's what you can do:
Take some 3/4" pipe insulation, sold for $5 at your local hardware retailer of choice.
Secure it with a couple of zip ties to the upper frame arm of your tire carrier.
No more rattle. Now I can hear myself think again while driving.