Does the function still need to be in memory to be loaded again ("does it just dump the pointer") or can I save it to disk, shut off the interpreter, boot it again and it imports it fine (in which case it somehow dumps them as code...?)?
Even in the linked test case on the readme you don't show the output/expectation of the serialization
Yes, it dumps them as bytecode (probably not compatible between completely different interpreters).
It even preserves debug metadata, so stack traces involving serialized/deserialized functions look right, and still show the original source file.
This is really neat.
I've shipped Love2D games as bytecode that wouldn't run on many Linux boxes because their LuaJIT installation (which is not part of Love2D but part of the system) was too old, or they stopped working after the user updated their system. There's a plethora of situations where something like that can happen.
I'm also wary of the "upvalues are preserved" feature, which sounds like a huge footgun, but I haven't looked into the details of your implementation.
I didn't include asserts in the linked case, because I thought it would be too verbose. You can see asserts in the test, that is linked below the example. Maybe it was the wrong call, I will think about including asserts into the example itself.
I think you could make it clearer, try reading the readme as someone with the preconceived notion that this is Yet Another Lua Serializer that translates functions, userdata and threads to their tostring() output. There are hundreds of those projects