Well, it turns out that the filtering is done by using whereDoesntHave, leaving the solution here for future reference.
$posts = Post::with('tags')->whereDoesntHave('tags', function ($q) {
$q->where('name', '=', 'is-project');
})->where('published_at', '!=', 'null')->orderBy('created_at', 'desc')->paginate(3);
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