Generally speaking, iterating over a container is likely to take about as much time as iterating over another, so if you keep adding to a container and then iterating over it, it's mainly a question of picking a container that avoids constantly having to reallocate memory and inserts the way you want quickly.
Both list and multimap will avoid having to reallocate themselves simply from adding an element (like you could get with a vector), so it's primarily a question of how long it takes to insert. Adding to the end of a list will be O(1) while adding to a multimap will be O(log n). However, the multimap will insert the elements in sorted order, while if you want to have the list be sorted, you're going to have to either sort the list in O(n log n) or insert the element in a sorted manner with something like lower_bound which would be O(n). In either case, it will be far worse (in the worst case at least) to use the list.
Generally, if you're maintaining a container in sorted order and continually adding to it rather than creating it and sorting it once, sets and maps are more efficient since they're designed to be sorted. Of course, as always, if you really care about performance, profiling your specific application and seeing which works better is what you need to do. However, in this case, I'd say that it's almost a guarantee that multimap will be faster (especially if you have very many elements at all).