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When Python is compiled with COUNT_ALLOCS, some tests in test_gc and test_module fail. I'm attaching the patch that skips 3 of them and modifies assertions in one of them, so that the tests pass.
I'm however still unsure about one of the skipped tests, since I'm unsure whether I totally understand what's wrong there - test_gc_ordinary_module_at_shutdown.
My guess is that due to COUNT_ALLOCS causing immortal types, the "final_a" and "final_b" types don't get destroyed on line [1] as they do in builds without COUNT_ALLOCS. AFAICS they are only "un-immortalized" on this line and destroyed during the following loop [2]. The problem here is that the order of destroyed modules is not deterministic, so sometimes the builtins module gets destroyed before the "final_X" and there is no "print" function, which makes the __del__ functions from "final_X" fail. IMO the best thing to do is to just skip this test with COUNT_ALLOCS. But I may be wrong, I don't have a great insight into Python's GC and module unloading.
[1] http://hg.python.org/cpython/annotate/0f48843652b1/Python/import.c#l383
[2] http://hg.python.org/cpython/annotate/0f48843652b1/Python/import.c#l394 

And the patch...

Since 3.4.0.b2, this also causes failures in another tests: test_io, test_logging, test_threading, test_warnings. There are various cases testing that some types get collected when the interpreter shuts down.
I'm attaching a new patch that covers all of these.

What is COUNT_ALLOCS?
Python 3.4 now has sys.getallocatedblocks() and a new tracemalloc module which are compiled by default.

As noted in Misc/SpecialBuilds:
COUNT_ALLOCS
------------
Each type object grows three new members:
 /* Number of times an object of this type was allocated. */
 int tp_allocs;
 /* Number of times an object of this type was deallocated. */
 int tp_frees;
 /* Highwater mark: the maximum value of tp_allocs - tp_frees so
 * far; or, IOW, the largest number of objects of this type alive at
 * the same time.
 */
 int tp_maxalloc;
...
We use this for Fedora's python debug build to get some interesting debugging info. (If you try to "grep -r" through Python 3.4 source code, you'll find quite a few ifdefs with COUNT_ALLOCS.)

Why do you check hasattr(sys, 'getrefcount') in test_io.py, but hasattr(sys, 'getcounts') in all other tests?

Good catch, using getrefcount was a mistake. I'm attaching a new version which always checks for getcounts (and also applies on 3.4.2).

LGTM (only one nitpick -- there are trailing spaces in test_gc).
But there are other tests which are failed with COUNT_ALLOCS. Here is a patch with additional fixes for these tests in test_gc and test_warnings.

Thenks Antoine for great idea proposed in comments on Rietveld. Following patch introduces strip_python_stdout() which strips COUNT_ALLOCS debug output from stdout (unfortunately this operation is not always unambiguous) and call it in assert_python_ok() and assert_python_failure(). This automatically fixes a large number of tests. Also fixed a number of other tests failing with COUNT_ALLOCS. Virtually all tests are now fixed except test_doctest and test_distutils.
A large part of the patch will be applied only to 3.4 and 2.7. With resolved issue23034 the patch for 3.5 will be much simpler, strip_python_stdout() and @requires_clean_stdout will gone away.

With Python-3.5 and COUNT_ALLOCS enabled following new tests fail also:
FAIL: test_is_finalizing (test.test_sys.SysModuleTest)
----------------------------------------------------------------------
Traceback (most recent call last):
 File "/builddir/build/BUILD/Python-3.5.0/Lib/test/test_sys.py", line 767, in test_is_finalizing
 self.assertEqual(stdout.rstrip(), b'True')
AssertionError: b'' != b'True'
---------------------------------
======================================================================
FAIL: test_print_traceback_at_exit (test.test_traceback.SyntaxTracebackCases)
----------------------------------------------------------------------
Traceback (most recent call last):
 File "/builddir/build/BUILD/Python-3.5.0/Lib/test/test_traceback.py", line 210, in test_print_traceback_at_exit
 self.assertEqual(stderr.splitlines(), expected)
AssertionError: Lists differ: [] != [b'Traceback (most recent call last):', b'[75 chars]ero']
Second list contains 3 additional elements.
First extra element 0:
b'Traceback (most recent call last):'
- []
+ [b'Traceback (most recent call last):',
+ b' File "<string>", line 8, in __init__',
+ b'ZeroDivisionError: division by zero']

COUNT_ALLOCS was added 22 years ago. I guess that the usecase is to track memory leaks, right?
branch: legacy-trunk
user: Sjoerd Mullender <sjoerd@acm.org>
date: Mon Oct 11 12:54:31 1993 +0000
files: Include/object.h Modules/arraymodule.c Modules/config.c.in Modules/imageop.c Modules/imgfile.c Objects/floatobject.c Objects/intobject.c Objects/listobj
description:
* Extended X interface: pixmap objects, colormap objects visual objects,
 image objects, and lots of new methods.
* Added counting of allocations and deallocations of builtin types if
 COUNT_ALLOCS is defined. Had to move calls to NEWREF down in some
 files.
* Bug fix in sorting lists.

00141-fix-tests_with_COUNT_ALLOCS-v5.patch: please don't do that! It makes tests much more verbose for a compilation option which is hidden and probably almost never used in the wild. The option has no configuration option for example.
*If* you really want to keep the feature, I would prefer to make it more visible (add a configuration option) and disable the output at exit by default. It's better to add a new "-X showcountallocs" as it was done with "-X showrefcount". Before, we had to fix a lot of unit tests (as 00141-fix-tests_with_COUNT_ALLOCS-v5.patch) to strip the "[xxx refs]" from stderr, it was very annoying.
"Python 3.4 now has sys.getallocatedblocks() and a new tracemalloc module which are compiled by default."
IMHO these two debug features superseded COUNT_ALLOCS. Please try to convince me of the use case of this very old debug feature.

FYI There is also issue23034 where is proposed "-X showalloccount" option to suppress the output, we use (custom patch) environment variable to filter out the verbose output in Fedora.

Yes, I also don't want to use 00141-fix-tests_with_COUNT_ALLOCS-v5.patch if there is better alternative. See issue23034 (I'm uncertain only in option name).

> IMHO these two debug features superseded COUNT_ALLOCS. Please try to convince me of the use case of this very old debug feature.
I no longer use this feature and I think that noone does. As you said, it seems to be obsoleted by other new features, so my vote would be to remove COUNT_ALLOCS altogether.

I propose to emit a compiler warning (or even an error?) in 3.5.x and drop
the code in 3.6. I don't think that a long deprecation period is requied.
The feature is not widely used.

New changeset 5abf6cdcac4d by Serhiy Storchaka in branch '3.5':
Issue #19527: Fixed tests with defined COUNT_ALLOCS.
https://hg.python.org/cpython/rev/5abf6cdcac4d
New changeset e7d84ecdd37d by Serhiy Storchaka in branch 'default':
Issue #19527: Fixed tests with defined COUNT_ALLOCS.
https://hg.python.org/cpython/rev/e7d84ecdd37d 

All tests now are passed in 3.6 on Linux. Making them passing in 3.5 requires too much changes that are not needed in 3.6. I don't think we need to pollute tests with these temporary workarounds.

New changeset 7b4ba62e388474e811268322b47f80d464933541 by Victor Stinner in branch '2.7':
[2.7] bpo-31692: Add PYTHONSHOWALLOCCOUNT env var (GH-3927)
https://github.com/python/cpython/commit/7b4ba62e388474e811268322b47f80d464933541