Heuristic for Finding Multiple Goals in Graph - e.g. using Kruskals Algorithm
MST is actually a tree, not a path (generally). Therefore, it is not a solution to TSP. (Although, MST can help you to find a solution that is not more than twice as long as the optimal solution, if you solve this in a metric space on a complete graph. That's unrelated story.)

Now, I must admit that I do not see a way spanning trees can help you if the maze contains loops. And in the case that the maze does not contain loops, it is somehow weird to talk about MST, because the graph of the maze is a tree.

However, maybe you can start to think about this: in a tree, for any two nodes, there is exactly one path that connects them. That's a trivial, but quite a nice property, isn't it? Length of this unique path is an easy lower bound that is more precise than Manhattan distance.

If you build MST over a graph with loops, this would probably not make an admissible heuristic, because the path using only MST edges can be longer than some path over edges you haven't choose to the spanning tree.

"Minimality" of MST is somehow a local property -- it's not like all the shortest paths are preserved. This is why I would never think about solving your problem with MST.