Since you are projecting, you are working with a new list and not with the original, LINQ is not a suitable tool for mutation - you also need a key to delete, not a value.
This should work:
var itemsToRemove = MemoryCache.Default
.Where( x=> x.Value is CachedObjectType &&
(x.Value as CachedObjectType).ZipCode == "11111")
.Select(x=> x.Key)
foreach(var key in itemsToRemove)
MemoryCache.Default.Remove(key);
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