Is it possible to get application fee refunded from PhD programs that miss admission decision deadlines? I'm sure I already know the answer to this question, but I feel compelled to ask.

With the passing of midnight a couple hours ago, the April 15 deadline used by many US graduate schools, by which applicants should accept/decline offers of admission (as decided by the Council of Graduate Schools) has passed. While I have made my choice, there is still at least one (pretty well-respected) school from which I never received a decision. Thus, in order to have waited out a decision from this program, I would have had to risk losing other offers (one of the conundrums that the April 15 resolution aims to avoid).

They were totally fine with taking my application fee back in November, but never seemed to get around to making a decision on my application (it's still pending online and there has been no response to emailed inquiries). I understand rolling admissions, but this is absurd. Is there any chance I will be able to get my application fee refunded, seeing as they did not keep their end of the deal? I guess I'm asking if anyone has ever successfully made this argument to a school.
You do not a have a leg to stand on, legally, morally or otherwise.  The thing you should actually indict them for is not responding to your emails. That is rude; I think you should probably accept that as an implicit rejection.  If they were still thinking of admitting you they would have checked on your status by this point.

The issue with your argument is that it doesn't actually follow the text of the resolution.  The binding part of resolution you cited says "Students are under no obligation to respond to offers of financial support prior to April 15."  So, schools have agreed not to withdraw an offer because you haven't accepted it before April 15th, and not to try to hold you to an offer you've accepted if you change your mind before April 15th.  The school in question has held to their end of the bargain perfectly, since they never made you an offer.  

The resolution says nothing about the school having to make a decision before April 15th. Of course, schools are well aware that it will be much harder to get people after April 15th, so they make an effort to make offers earlier. However, this is not a deal, this is the department acting in their own self-interest.  

Unfortunately, there's too much uncertainty in the process to guarantee you'll make all offers before April 15th.  After all, you have some plan for what your incoming class will be, you make offers to many more people than that, and you hope that you guessed the yield correctly.  If the people who've already been offered admission are waiting until April 15th to make up their minds, you're then left with a great deal of uncertainty as the deadline approaches, since it will be a big problem if your incoming class is such bigger or much smaller than you planned.  Thus, there often are people who you are not sure you want to reject, but not sure you have the money to fund.  In my department, we still have a number of people waitlisted, since we haven't filled our class yet.  Usually, such people will be formally notified they are on a waitlist; I think again, that's in the department's self-interest and helps clarify things for the student.  So, again, it's likely this is an implicit rejection, which for whatever reason they haven't bothered to formalize.