It's unclear from the FCC's website and 47 CFR part 87 (Aviation Services) whether or not an address update is required for holders of a Restricted Radio Telephone Operators Permit (RR). I've never heard of any requirement to do it either. There are requirements for Amateur Radio Operators to do so, and it wouldn't hurt anything if you were to update the FCC to your current address.
The page where it explains how to do this update via the FCC's Universal Licensing System website can be found here. It also states that there is no fee charged for doing so. I can find no requirement that once you update the address that you must have a license with that updated address on your person (its the same way with FAA certificates). Though if you do update the address you should be able to print a new license from the FCC's website with the updated address. Hope this helps.
I've had a FCC General Radiotelephone License since 1970. There is no requirement to update your address with the FCC. I've moved many times since then. You just need to be able to present your paper license if required.