There is no one-sized-fits-all application, so an MP won't be able to help in each case.
The basic thing to remember is that while talking to an MP *might* help your application, it will *never* hurt your application. There is NO risk involved. Only possible benefit.
So, for those who say "I've heard it's useless so don't bother," fine, okay. It was useless for someone before. But occasionally it helps people. Yes, it stinks that there is no magic bullet for EVERYONE, but that doesn't mean that all effort is useless.
The cases that an MP is most likely to help in are ones like Dan11's. Ones where CIC has *forgotten* to inform the client of something. Like needed fingerprints. Another thing that the MP can do is inform you of your status. CIC does not inform applicants if they are scheduled for secondary review or citizenship judge hearings (not sure if they're even still doing that...) Your MP can tell you. CIC is obligated to respond to MP's inquiries with an update, especially for applications beyond typical processing timelines.
The party of the MP makes no difference. One of their jobs is to be the "ombudsman" of their constituents for the federal government. They are there to advocate for you. If they don't do so, I encourage you not to vote for them once you get your citizenship.
And, no, you can't just go to any random MP. You can try, but MPs are responsive to their constituents (people who live in their riding) not other people's constituents.
Go to your MP.
Good luck.