No, the majority of the cache owners do not stipulate if a permit is required or not.
(keep in mind this is a personal opinion that some do not agree with)
As a cacher and a responsible outdoor enthusiast, it is our responsibility to make sure we do not break the rules of the land managers. As such all caches in these areas ^Should^ have a notice or information, but it is not in any way enforced or required by Groundspeak.
Apache-Sitgreaves : No permits required that I know of
So, without mentioning State Parks or any other private ownership areas, these are the ones I can think of that require some form of permit or contact. I am sure I missed a few, but heck, it is information gained at no time or cost to you. :)
ShadowAce / Dirk