If you are editing a PHP file and want it to auto-complete the HTML tags, in addition to turning on "Auto-insert HTML/XML tags" in Preferences, you have to change the programming language to HTML while editing that file.
In Notepad++, go to the Language menu, and find HTML and click on it.
This will set the file's language to HTML so that auto-complete tags will work in a PHP file.
Even with the language set to HTML, it still highlights anything in tags as PHP code, and it does not affect auto-closing of brackets, if you have that enabled.
You must have Auto-Insert html/xml close tags on, as mentioned in Noha's answer.