Sort of.  Consider the tree in my example graph:

A -1- B -1- C -1- D -1- E -9- J
  -2- F -1- C -1- D -1- E -9- J
        -2- G -1- D -1- E -9- J
              -2- H -1- E -9- J
                    -2- I -1- J

There's no circling going on as you depth-first search this tree, even though you are wasting time visiting the same node multiple times.

However, the thing you know with A*/djikstra is this: If I have visited a node, there is no shorter path to that node.  So any time I encounter that node again, I must have at least as long of a path, and so any later nodes along that path can't be any better along this path.

Effectively, you are pruning the tree:
A -1- B -1- C -1- D -1- E -9- J ***
  -2- F -1- C ***
        -2- G -1- D ***
              -2- H -1- E ***
                    -2- I -1- J GOAL
(*** = pruned branches)

since the second time you visit C, you know the first path was faster, so there is no reason to continue to visit D/E again.  This is even more noticable in graphs with multidirectional edges, as the tree is infinitely deep at every cycle.