The left hand rule assumes that the end is on the same "cycle" as the entrance:

   --E--
  |     |
  |     |
   --S--

if the end is instead on a different "cycle", it won't work:

   -------
  |       |
  | -- -- |
  ||     ||
  ||  E  ||
  | ----- |
  |       |
   ---S---

Hopefully this makes sense. I'm sure there's a propper phrasing of this in topology, but I don't know it.

"Multiply connected" is the topological term used in the English-language literature on mazes.

A maze where the goal is not on the edge of the maze. In this case, the goal is in the center.

3-dimensional mazes (which have bridges or tunnels) also often cannot be solved with this rule.

You first statement is not correct. The rule is perfectly useful to solve mazes where the goal is in the center.

The requirement is that the walls should be connected, i.e. that there are no free-standing walls somewhere in the maze not connected to the border walls, for example.

I stand corrected! I suppose it is possible to have a center goal that is connected to the edge by a wall.

Any maze with a wall that loops around itself.

Yes, in a maze without cycles the left-hand rule will take you through every part of the maze, guaranteeing that you reach the goal.

Of course in a maze with cycles you might still reach the goal, but it's not guaranteed. For example there's a cycle in Hampton Court Maze, but the left- and right-hand rules both get you to the goal from the entrance.

They are referred to as "multiply connected" in the literature on mazes. Essentially, this is the mathematical way of saying that there are "islands" of hedge, not attached to the perimeter.