With regards to the original question - I don't have any good ideas, probably valuation rings satisfy this (but they are not generally Noetherian except in the cases already outlined).
With regards to other classes of ideals that satisfy this (now in characteristic zero), the ideals of ``unions of log canonical centers'' satisfy this property in characteristic zero (this is mostly due to a result of Florin Ambro I think). See for example this preprint.
