I have a result from my Phd that I've been trying to get published somewhere. I don't know if I should ask either my Phd or PostDoc supervisor for advice (In theoretical computer science / math). I am a bit reluctant because I'm not sure to what extent your PI is an advisor as a postdoc especially for self publication and about the breakdown of group projects versus personal projects during a post doc (although I did all the research during my phd).
If you mean publishing as a sole author, yes, you can ask and it might be good to get their advice on whether your work is ready for "prime time". You aren't required to get that advice, however, if the work is yours alone. If you submit your work then you will get feedback from editors and reviewers. But, advice is good when it comes from people you trust.
If you mean publishing on a personal website or otherwise not involving a journal or conference, again, you don't need their advice, and you can do as you please with your own work. But, self-publishing in this sense won't advance your career much in the early stages. It will be difficult for the right people to find your work. Once you are established it isn't such a big issue if the people who want to read what you write already know how to find you.