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I would like to see this posted as a recipe before being put in the standard library. It needs a chance to mature and to demonstrate that people will want to use it.
FWIW, the example is problematic in a couple of ways. The registration of shutil.rmtree(tmpdir) will run *before* mngr register is called. 
Also, it doesn't take advantage of any of the with-statement features. It doesn't show any advantage over a standard try/finally which is arguably cleaner:
 tmpdir = tempfile.mkdtemp()
 try:
 # do stuff with tmpdir
 finally:
 shutil.rmtree()
Also, I suspect that the CleanupManager would be an error-prone construct because the registration occurs somewhere after the with-statement is set-up, possibly resulting in errors if there is an early, pre-registration failure.