SO does not hide the posts (AFAIK), it just highlights or unhighlights them. And I am sure they are basically using CSS classes for that.
From (very) quick look at the source, they basically use tags as CSS classes and then load a unique-to-you javascript that adds styles to entries with those classes.
You basically would need to generate a visitor specific CSS or Javascript which you generate based on the tags. Plus user interface to select/unselect those tags (I don’t like the one SO uses). Plus extending WordPress user model to store those preferences.
Not a big project, but a medium size one.