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regrtest already tests for refcount leaks and memory allocation leaks. It can also be made to check for file descriptor leaks (and perhaps also handles on Windows).
Running with the attached patch makes it look like test_openpty, test_shutil, test_subprocess, test_uuid all leak fds on Linux, but I have not investigated:
$ ./python -m test.regrtest -R 3:3 test_openpty test_shutil test_subprocess test_uuid
[1/4] test_openpty
123456
......
test_openpty leaked [2, 2, 2] fds, sum=6
[2/4/1] test_shutil
beginning 6 repetitions
123456
......
test_shutil leaked [4, 4, 4] fds, sum=12
[3/4/2] test_subprocess
beginning 6 repetitions
123456
......
test_subprocess leaked [5, 5, 5] fds, sum=15
[4/4/3] test_uuid
beginning 6 repetitions
123456
......
test_uuid leaked [1, 1, 1] fds, sum=3
4 tests failed:
 test_openpty test_shutil test_subprocess test_uuid

New changeset a7381fe515e8 by Richard Oudkerk in branch '2.7':
Issue #18174: Fix fd leaks in tests.
http://hg.python.org/cpython/rev/a7381fe515e8
New changeset 46fe1bb0723c by Richard Oudkerk in branch '3.3':
Issue #18174: Fix fd leaks in tests.
http://hg.python.org/cpython/rev/46fe1bb0723c 

The test_shutil leak is caused by #17899. The others are fixed by a7381fe515e8 and 46fe1bb0723c.

Updated version which adds checks for handle leaks on Windows.

Just flagging this up as testing is rather important.

The patch looks good.
I just think it would be nice to expose _fdcount() in test.support.

Why not using os.fstat() instead of os.dup() to check if a file descriptor is open or not?
It's strange to create a new file descriptor with os.dup() to count the file descriptors.

> Why not using os.fstat() instead of os.dup() to check if a file descriptor is open or not?
I asked myself the same question, but IIRC, fstat() doesn't always
work on Windows (of course).

> I asked myself the same question, but IIRC, fstat() doesn't always
work on Windows (of course).
Can you please elaborate?

> STINNER Victor added the comment:
>
>> I asked myself the same question, but IIRC, fstat() doesn't always
> work on Windows (of course).
>
> Can you please elaborate?
Not really, since I don't know much about Windows, but that's
something I think I heard.
Richard will be able to give more details.

I can't remember why I did not use fstat() -- probably it did not occur to me.

Modified patch to use os.fstat(), and try to use /proc/self/fd/ on Linux.

New changeset 379aad232000 by Victor Stinner in branch '3.4':
Issue #11453, #18174: Fix leak of file descriptor in test_asyncore
http://hg.python.org/cpython/rev/379aad232000
New changeset 0ced2d2325fb by Victor Stinner in branch 'default':
(Merge 3.4) Issue #11453, #18174: Fix leak of file descriptor in test_asyncore
http://hg.python.org/cpython/rev/0ced2d2325fb 

New changeset 746339776f19 by Victor Stinner in branch '3.4':
Issue #18174: Fix leak of file descriptor in test_tempfile
http://hg.python.org/cpython/rev/746339776f19
New changeset 017d701116d5 by Victor Stinner in branch 'default':
(Merge 3.4) Issue #18174: Fix leak of file descriptor in test_tempfile
http://hg.python.org/cpython/rev/017d701116d5 

> Richard Oudkerk added the comment:
>
> I can't remember why I did not use fstat() -- probably it did not occur to me.
I probably have Alzeihmer, I was sure I heard a reasonable case for
dup() vs fstat().
The only thing I can think of is that fstat() can incur I/O, whereas
dup() shouldn't.
In any case, I too prefer fstat(), since it doesn't require creating a
new FD (which could fail with EMFILE/ENFILE), and is more logical.
The only comment I have about this patch is I think that the helper
function to detect the number of open FD might have its place in
test.support.

fstat() can do I/O, or can fail for other reasons. If you don't want to create a new fd, I think you can do dup2(fd, fd).
I don't understand the reason for the following code:
+ def check_handle_deltas(deltas):
+ return abs(sum(deltas)) >= min(3, len(deltas))
Can you add a comment?

> fstat() can do I/O, or can fail for other reasons.
Oh, I forgot this reason. Maybe we should add a comment to explain that. I mean in the patch for this issue but also in is_valid_fd() (Python/pythonrun.c).

New changeset 0e7d71a3bf0d by Victor Stinner in branch 'default':
Issue #18174: Explain why is_valid_fd() uses dup() instead of fstat()
https://hg.python.org/cpython/rev/0e7d71a3bf0d 

New changeset 72129c767c92 by Victor Stinner in branch 'default':
Issue #18174: "python -m test --huntrleaks ..." now also checks for leak of
https://hg.python.org/cpython/rev/72129c767c92 

I commited the first part to check for file descriptor leak.
I didn't commit the second part, to check for Windows handle leak.

New changeset ec2ef7525fa5 by Victor Stinner in branch 'default':
Issue #18174: Fix test_regrtest when Python is compiled in release mode
https://hg.python.org/cpython/rev/ec2ef7525fa5 

New changeset 270581905cab2747ae8f7ee945301d6a29509cc7 by Victor Stinner in branch '2.7':
bpo-18174: Fix file descriptor leaks in tests (GH-7408)
https://github.com/python/cpython/commit/270581905cab2747ae8f7ee945301d6a29509cc7

New changeset 64856ad8b7279718ff10a9fb32003c2221af7228 by Victor Stinner in branch '2.7':
bpo-18174: regrtest -R 3:3 now also detects FD leak (#7409)
https://github.com/python/cpython/commit/64856ad8b7279718ff10a9fb32003c2221af7228

New changeset bc3df70b266304f78ebe5eabead71cabd4738d12 by Victor Stinner in branch '2.7':
bpo-18174: Fix fd_count() on FreeBSD (GH-7420)
https://github.com/python/cpython/commit/bc3df70b266304f78ebe5eabead71cabd4738d12

I give up on that close. I closed my PR 7827 and PR 7966.