I generally chase after whatever my adviser/supervisor tells me to do, although I know that it's sometimes a project that they don't want to spend their time on. I know that I can learn a lot even when I go after such "wild goose chases".
I recently talked to a PhD student though, and he advised me that I should learn when to say "no" to an adviser, to recognize that sometimes those suggestions can lead to "wild goose chases" that aren't worth the time.
What are some guidelines when an adviser wants you to chase a problem that might not be worth its time investment?
This is a difficult issue to deal with. You are correct in stating that (in some disciplines) this can be a significant problem, but as a graduate student, it may be hard for you to argue your case that the project will lead to a dead end.
I see two possible solutions:
If you have a good relationship with your advisor, speak with him about your concerns. He may admit that he's not sure where the project will lead, but he will likely be willing to give you the background as to why he's interested in the project, and where it will lead you. These types of projects are also good opportunities to ask your advisor to introduce you to collaborators with whom you can complete the project, as he's only tangentially interested.
If your advisor likely won't listen to you, then put in the month or three to do the necessary background research to prove your case. Look up the references, research previous findings, contact others who have worked in the field. One of two things will happen: (1) you will change your mind, or (2) you'll build a strong case to present to your advisor as to why this research is not worth yours or his time. If at that point he still wants you to work on the project, then either he has political motivations or he's just being unreasonable, both of which are indicative of larger problems which you should deal with.