Does this do what you want?
static ILookup<TKey, TValue> Copy<TKey, TValue>(ILookup<TKey, TValue> lookup)
{
    return lookup.
           SelectMany(g => g,
                     (g, v) => new KeyValuePair<TKey, TValue>(g.Key, v)).
           ToLookup(kvp => kvp.Key, kvp => kvp.Value);
}
Of course, if you want to transform the values somehow, maybe you want something like this:
static ILookup<TKey, TValueOut> Transform<TKey, TValue, TValueOut>(
       ILookup<TKey, TValue> lookup,
       Func<TValue, TValueOut> selector)
{
    return lookup.
           SelectMany(g => g,
                      (g, v) => new KeyValuePair<TKey, TValueOut>(g.Key, selector(v))).
           ToLookup(kvp => kvp.Key, kvp => kvp.Value);
}
Note that this method holds intermediate values in a KeyValuePair which, being a value type, is stored on the stack and thus doesn't require any intermediate memory allocations. I profiled a test that creates a Lookup<int,int> with 100 keys, each having 10,000 items (for a total of 1,000,000). 
- Creating the 
Lookup does 1610 allocations. 
- Copying it with my method does 1712 allocations (all the allocations required to create it plus one for each delegate in the 
SelectMany call and one for the enumerator for each key). 
- Copying it with anonymous objects instead of 
KeyValuePair does 1,001,712 allocations (all the allocations required to copy plus one for each item). 
CPU-wise, even with 100,000 elements per key in the Lookup performance between the two copying methods was identical. With 1,000,000 elements per key, the performance was different between the two methods:
- 5.1 sec to create
 
- 5.9 sec to copy with 
KeyValuePair 
- 6.3 sec to copy with anonymous objects