Do I need permission from my past supervisor?
If your past supervisor made a significant intellectual contribution to the new paper, then he should be a coauthor.  And then, yes, you need his permission to publish the paper, because publication requires the consent of all coauthors.

If your past supervisor did not make a significant intellectual contribution to the new paper, then he should not be a coauthor.  And then no, you don't need his permission to publish the paper, because he is not a coauthor.

In short: The fact that he is your former supervisor is utterly irrelevant.  (This would also be true if he were your current supervisor.)