Is the chess problem a dead form? Or, to ask the question a bit less opinionatedly, "what new ideas have been introduced in chess problems over the last 20 years?"  I imagine many problems have been composed in that time, but it seems like all of the stipulations (model mates, selfmates, etc.), all of the structural elements (pinnings and unpinnings, interference, and so on) and all of the tasks (Phoenix tasks, etc) have been long since sussed out.  The only relatively recent development that I'm aware of are various 'Babson cycles' that have been composed by Peter Hoffmann (see Wikipedia's page on the Babson task for details).  Are there any other relatively recent developments of note in chess problems?
An unsorted list of arguments for the vitality of chess problems (probably my answer is a little bit biased because I am an enthusiastic chess composer ...) 


Computers: A lot of new ideas have been developed during the last decades. One reason for it is the rise of computer programs which help to check the correctness of a chess problem. This simplified the chess composition dramatically, especially selfmates and helpmates with very complex and long solutions are now possible...
Helpmates: by the way help mates. This is an area I know a little bit and during the last 20 years there was phantastic progress. Here some non representative samples
http://kobulchess.com/en/problems/originals2013/321-chess-problem-helpmate-viktoras-paliulionis.html 
or http://www.yacpdb.org/#320933 
or http://pdb.dieschwalbe.de/search.jsp. Study the solutions and look for other problems of these authors (and other like Caillaud, Mazlar ...) to find other good help mates ... also there are still  many open themes which are not realised. such as the '100$ theme' , the 'Oudot theme' or a helpmate with more than 27 moves...
Specialisation:  Chess compositions branch in many different ways. Some "theoretical" knowledge on certain themes is restricted only to a very small group of composers. So for example if you ask for the development of some themes for three movers (mate in three) the current state of the art is probably known only to a very small group of composers (probably less than 100 people world wide). I am definitely sure they know exactly how their field developed during the last years. And definitely they will argue for a good progress in their field of work. But from "outside" the progress seems very modest (of course I don't want to blame the three mover community but this is an area I know almost nothing  :-)... There is a similar situation in mathematics or other scientific areas where the current state of development is often only understood by a small group of experts...  
Fairy chess: If you think of chess problems in the broader sense of generalized chess the we come to fairy chess. This is the genre where the chess rules and the stipulations are extended far beyond the orthodox setup. For example new varieties of chess (varieties of circe chess, ghost chess ,... et.al.) and many many different pieces have been developed during the last decades and all of them help to represent new ideas in chess problems (or can you think of triple check or ...). For much more on fairy chess look at juliasfairies.com
selfmate, retro problems ...: There are a lot of other areas where chess problems develop. For example look at  the (german) retro blog http://www.thbrand.de/retroblog/ where you can learn a lot about retro chess (if you are aware of german)...


I hope all this helps to believe in the vital functions. For much more infos check the websites I mentioned above