There are two map tiles next to each other, but separated by the edges. Meaning there is a black line between those spaces. The rulebook says that normally figures can't move between the black lines. Can I use the Teleportation Rune to go from one square to the other? Can I teleport through the black edge?

No. Even when teleporting you need to follow rules for counting spaces. You cannot count through obstacle terrain, black edges, closed doors or locked doors, or gaps in map tiles.

EDIT: Examples of things you can count through that you can't move through:

portcullis

elevation line

non-friendly figure

object that "blocks movement"

You can also count through pit spaces and water spaces (any terrain other than an obstacle space) as if it were a normal space.

Teleportation isn't moving though, it is placing. So why would you not be able to count through a closed door for example?

EDIT: Ah ok we're talking adjacent spaces and counting adjacent spaces. Makes sense then.

Teleportation isn't moving though, it is placing. So why would you not be able to count through a closed door for example?

EDIT: Ah ok we're talking adjacent spaces and counting adjacent spaces. Makes sense then.

Correct- there needs to be a continuous line of adjacent spaces from where you start to where you end. Since spaces on the other side of doors/black walls/obstacles are not adjacent, you cannot count across such things- they break your chain of adjacent spaces. On the other hand, elevation lines, portcullises, and figures do not interfere with adjacency. This is the key difference.