Ownership is doable in a dynamic language, provided you are willing to tolerate aborts: C++'s std::unique_ptr is enforced at runtime. Similarly, you can emulate dynamic borrowing using something like Rust's RefCell, which aborts when you attempt to mutably borrow a value twice (though it can be quite difficult to reason about from a usability perspective).
But if you're building on top of F# you have a static type system at your disposal. Usually, languages that incorporate linear types and first-class garbage collection treat linear types as a separate kind, and allow them to interact with standard types only in limited ways. See, for example, https://github.com/idris-lang/Idris-dev/wiki/Uniqueness-Type.... I believe http://protz.github.io/mezzo/ is also exploring this design space.
My lang will have a static type, so that info is usefull.
It will start as a interpreter, but probably the challenge is the ergonomics of it: How make it "easier" to use... probably limit it to only handle resources? (ie: files, handles, etc, like the IDisposable in .net)
But if you're building on top of F# you have a static type system at your disposal. Usually, languages that incorporate linear types and first-class garbage collection treat linear types as a separate kind, and allow them to interact with standard types only in limited ways. See, for example, https://github.com/idris-lang/Idris-dev/wiki/Uniqueness-Type.... I believe http://protz.github.io/mezzo/ is also exploring this design space.