Question: As they say, "politics is the exercise of power", and "power is the ability to make things happen in an organisation".
I am working on a project which has some overlap and is partially in competition with other projects. Managers at all levels have different ideas about what should happen. I am trying to understand who is supporting whom, what are the networks I should be aware of, to have a clear idea of where support for our idea might come from.
My line manager doesn't want me to think about politics at all. Other people told me not to be political. But I think these people might have their own agendas - like everybody else. I have seen what happens when you work hard on a project, ignore the politics, and then get backstabbed by some unforeseen person or project.

I do believe that "ignore politics" is bullshit advice: whether you manage people or build things, politics will determine the allocation of resources to your team or project, but I might be wrong.
So, let's ask it this way: **"what are the circumstances when ignoring company politics is good advice?"**

Answer: 
Even as an outside contractor or consultant, I have to agree broadly with Philipp that it is rarely a good idea to ignore politics.

That isn't to say you should spend your time "stirring the pot", or getting into debates and complex discussions. But, I think your premise is right - you need to be aware of what is going on, who the decision-makers are, what controversies there are and so on.

Additionally, you always owe it to yourself to protect yourself. This means ensuring you are credited for your work, ensuring that people are aware of your contribution and ensuring you don't get blamed for others mistakes.

To that end, I don't think any professional who is looking to progress their career can "ignore politics"
Your manager telling you to not think about politics is a political act. Which means he's told you not to think about that advice. You can't follow advice without thinking about it.

Nonsense is never good advice. Ignore this particular advice but don't be obvious about it.