Can I use MongoDB as a replacement for CoreData on iOS?

MongoDB is probably not a good fit here. The server assumes that disk space is not a limiting factor. The current stabel brance (1.6.x) does not provide data durability on a single instance. It also uses memory mapped files which are fine on dedicated servers but will cause lots of disk churn and unnecessary memory use on a typical PC. I have no experience with Core Data, but it sounds like a septate niche to me.

MongoDB is not designed to be used as an embedded database. It is designed for speed and scalability and unlikely fits the architecture of an embedded OS like IOS. However talking to some MongoDB-backed service in the "cloud" through standard protocols like HTTP/S should not be an issue. You just need to write some web-gateway in front of MongoDB.

As an aside, I will note that it's a common misperception but Core Data is not a database system.

Instead, it is a runtime object graph management system with persistent tacked on as option if you want it. It's primary function is to provide the model layer of the Model-View-Controller design pattern. As such, it deals with a lot more than just getting data on and off a disk.

Core Data does have a learning curve but in my experience the biggest obstacle many face is trying to treat Core Data as some kind of object oriented wrapper around SQL. From that perspective Core Data is very confusing because it seems to require to learn so much that has nothing to do with persistence.

Using database like SQLite, MongoDB and CouchDB for persistence won't really speed things along at all because, although you might better understand how they get data on and off the disk, they won't help at all in managing the data and the relationship to the other data objects and the objects of the UI. You still have to have a data model and you will have to code all that up by hand. Unless your data model is extremely simple, that will take more time than learning Core Data.

The best way to learn Core Data is to ignore the fact that the object graph can be persisted at all. Just start from the perspective that you've got a lot of objects that represent the data model of you app and you have to manage their attributes and relationships.