When you do dependency injection well, you don't inject every object; that would be horrible/impossible. What you may have noticed is that there are two kinds of objects, ones that you inject and ones that you don't. The ones you don't are things like numbers, strings and maps/sets; things that you treat as values. The other objects do things, I tend to call them services.

In order to do a straight-forward conversion to functional programming, I suggest leaving the values as they are and each service becomes a free monad transformer. So, instead of having a logger, you have a logging monad transformer that has a log instruction. Instead of having a database, you have a database monad transformer that has a query instruction, etc.

You are then free (no pun intended) to replace the interpreters of these free monads during testing with whatever mock implementation you please and the result is a more principled dependency injection inspired style.

Actually, I would constrain the monad type via type classes, rather than using free monads, but the approaches are equivalent.