Can I instigate collaboration outside of my current research team, or is that bad etiquette? I recently finished my PhD and moved on to a postdoc research position in a semi-related subject area at a different University. I have an idea for a narritive review in my old subject area and I have some time to make a start on it before the current project gets really busy. 

My issue is that I would like to invite my old PhD supervisor and another researcher from a third University, who we have collaborated with before, to co-author the paper with me as they are experts in the area that the review would cover. My current postdoc supervisor/line manager and other research colleague do not have any expertise in this subject area. 

Is it ok to start writing the paper with my old colleagues without permission or notification of my current research team/boss? Or should I tell them that I am writing a paper with my old colleagues, without being able to offer them the chance to be involved? Not sure what the expectation is and don't want to annoy anyone at my (very new) job.
You should not start working on an independent project without telling your current supervisor. After all, he acquired the funding which pays your salary, so he needs to be on board with everything you do that takes up a significant amount of time. (Of course, I am not talking about asking for his permission every time you review a paper.)

Then again, as a postdoc (as compared to Ph.D. students), you are expected to start spreading your wings and collaborating with people outside your supervisor's immediate group. So chances are that your supervisor will be supportive, especially since you write that you have some time to kill.

But ask first; it's possible that your supervisor knows something that needs to be done before your project "really" starts.

This question is related (full disclosure: I answered there): Should postdocs work only on the projects they are hired for, or can they work with other people in the same research group on other projects?